Term for a set with no limit points?
Angus Rodgers
I'm just checking some elementary terminology in general topology,
as there doesn't seem to be a simple name for a simple concept.
First, the following terms all seem to be synonymous, in the sense
that although different authors chose different terms from the list,
no author also uses any of the other terms in a different sense:
limit point
accumulation point
cluster point
Is that right?
Second, the set of all limit points (accumulation points, cluster
points) of a subset A of a topological space X is always called
the "derived set" of A. It is usually denoted by A'; however,
G. F. Simmons, in /Introduction to Topology and Modern Analysis/,
uses D(A) (having already used A' for the complement of A).
Third, a subset A of a topological space X is usually called
"discrete" iff it forms a discrete space when given the subspace
topology. Exercise 4G in Willard, /General Topology/, ought to
be a sufficient reference for this. (I got tired of searching at
this point.)
However (and this brings me to the point at issue), Narasimhan
and Nievergelt, in /Complex Analysis in One Variable/ (2nd ed.),
Theorem 2 of section 1.3, define a subset Z of a connected open
set D in C to be "discrete" iff it has only isolated points and
is closed (as a subset of D, rather than C - although they don't
say this explicitly).
As I learned in the thread "Zeros of an analytic function", this
property characterises those subsets Z of a domain D which are
the zero sets of analytic functions D -> C.
In passing, let me check also that the terms "domain" and "region"
are also synonymous, as both meaning an open, connected subset of
C (which may also be required to be non-empty - but I won't worry
too much about this optional extra condition). Is that right?
I think the usage of Narasimhan and Nievergelt must be rejected
(in spite of their authority), but what does this leave us with?
Rudin, in /Real and Complex Analysis/ (3rd ed.), simply describes
the characteristic property of zero sets of analytic functions
D -> C as that of having no limit points in C. This is simple
enough, I suppose, and better than my clumsier description of
them, in that other thread, as "(i) discrete, and (ii) closed
as a subset of D (but not necessarily as a subset of C)".
But is there not a simpler term - somewhere! - for a subset of a
topological space whose derived set is empty? I can't find any.
("Type zero" was the only suggestion I came across - I forget
where - and "scattered" also suggests itself, but seems to mean
something not quite the same, although closely related.)
--
Angus Rodgers
Angus Rodgers
>Rudin, in /Real and Complex Analysis/ (3rd ed.), simply describes
>the characteristic property of zero sets of analytic functions
>D -> C as that of having no limit points in C.
Fortunately the typo is obvious: that last C should be D, of course.
--
Angus Rodgers
Dave L. Renfro
> Rudin, in /Real and Complex Analysis/ (3rd ed.), simply describes
> the characteristic property of zero sets of analytic functions
> D -> C as that of having no limit points in C. This is simple
> enough, I suppose, and better than my clumsier description of
> them, in that other thread, as "(i) discrete, and (ii) closed
> as a subset of D (but not necessarily as a subset of C)".
>
> But is there not a simpler term - somewhere! - for a subset of a
> topological space whose derived set is empty? I can't find any.
>
> ("Type zero" was the only suggestion I came across - I forget
> where - and "scattered" also suggests itself, but seems to mean
> something not quite the same, although closely related.)
Note that having no limit points in C is equivalent to having
a finite intersection with every disk (open or closed) centered
at the origin, or equivalently having finite intersection with
every disk (open or closed) in C, or equivalently having finite
intersection with every bounded subset of C. It might be tempting
to say this is also equivalent to being a uniformly isolated set
(this means there exists a positive lower bound on all possible
distances between distinct points in the set), but the example
{n + 1/n: n = 2, 3, 4, ...} shows (half of the statement) that
being uniformly isolated is strictly stronger. Another useful
fact is that in R^n (I'm not sure to what extent this holds
in more general topological spaces) a set is isolated if and
only if it has empty intersection with its derived set.
Note that if the derived set is empty, then the set certainly
has this property. On the other hand, the midpoints of the
bounded complementary intervals of any Cantor set in R form
an isolated set whose derived set is the Cantor set union
those midpoints. Thus, while having an emtpy derived set
implies the set is isolated, a set can be isolated and yet
still have a fairly large derived set (one that has positive
Lebesgue measure, for instance). It's always the case that
the derived set of an isolated set is nowhere dense, though.
[Why? Well, every derived set is a closed set. Thus, if the
derived set were dense in some neighborhood, then the derived
set would in fact fill up some neighborhood, and no isolated
set has this property.]
I don't know a name for a set whose first derived set is empty,
although this seems like something I ought to know. However,
all my books and other references are at home, and I'm not,
so I'll pass on this. Places I'd look if I had access to my
stuff are R. Vaidyanathaswamy's book "Set Topology" and
K. Kuratowski's "Topology" (1966 English edition, Volume 1).
Vaidyanathaswamy's book
http://books.google.com/books?id=yDMipybQ64kC
Scattered sets are sets that become emtpy after some countable
iteration of the derived set operation (take intersection
of all previous derived sets at the limit ordinal stages).
I've posted a lot about these notions (isolated sets, the
derived set operator, scattered sets, etc.), but the following
posts seem to be the most relevant to what you're asking about.
sci.math -- unique limit point in infinite bounded set
(Dave L. Renfro; 13 & 17 June 2007)
http://groups.google.com/group/sci.math/msg/23e6146526f92a98
http://groups.google.com/group/sci.math/msg/1320c00bd89012ff
sci.math -- Distrubutive Property of Limit Points
(Dave L. Renfro; 10 October 2005)
http://groups.google.com/group/sci.math/msg/3727727671de641e
sci.math -- Question from Rudin
(Dave L. Renfro; 15 May 2000)
http://groups.google.com/group/sci.math/msg/a31647549c87f4cf
sci.math -- MONOTONE SET FUNCTIONS, FIXED POINTS,
AND CECH CLOSURE FUNCTIONS
(Dave L. Renfro; 4 July 2005)
http://groups.google.com/group/sci.math/msg/9eb76fc26a4cacff
http://groups.google.com/group/sci.math/msg/550f93d6f4f252ed
Dave L. Renfro
Dave L. Renfro
> [Why? Well, every derived set is a closed set. Thus, if the
> derived set were dense in some neighborhood, then the derived
> set would in fact fill up some neighborhood, and no isolated
> set has this property.]
Although I wrote this correctly, just to make sure there's
no misunderstanding, by "no isolated set has this property",
I mean the property that its derived set fills up some
neighborhood. One way to see this that gives some additional
relations among these notions is to note that every isolated
set is nowhere dense and the closure of a nowhere dense set
is nowhere dense. Since the closure of a set is the set
union its derived set, it follows that the derived set
is nowhere dense (any subset of a nowhere dense set is
nowhere dense, and the derived set is a subset of the
closure of the set).
Dave L. Renfro
Angus Rodgers
<renf...@cmich.edu> wrote:
>Angus Rodgers wrote (in part):
>
>> Rudin, in /Real and Complex Analysis/ (3rd ed.), simply describes
>> the characteristic property of zero sets of analytic functions
>> D -> C as that of having no limit points in C.
("C" corrected to "D" in followup.)
>> This is simple
>> enough, I suppose, and better than my clumsier description of
>> them, in that other thread, as "(i) discrete, and (ii) closed
>> as a subset of D (but not necessarily as a subset of C)".
>>
>> But is there not a simpler term - somewhere! - for a subset of a
>> topological space whose derived set is empty? I can't find any.
>>
>> ("Type zero" was the only suggestion I came across - I forget
>> where - and "scattered" also suggests itself, but seems to mean
>> something not quite the same, although closely related.)
>
>[...] It's always the case that
>the derived set of an isolated set is nowhere dense, though.
>[Why? Well, every derived set is a closed set. Thus, if the
>derived set were dense in some neighborhood, then the derived
>set would in fact fill up some neighborhood, and no isolated
>set has this property.]
(I'll read your followup before going through this paragraph.)
>I don't know a name for a set whose first derived set is empty,
>although this seems like something I ought to know. However,
>all my books and other references are at home, and I'm not,
>so I'll pass on this. Places I'd look if I had access to my
>stuff are R. Vaidyanathaswamy's book "Set Topology" and
>K. Kuratowski's "Topology" (1966 English edition, Volume 1).
By moving a chair out of the room, borrowing a stool from
another room, and moving three rickety piles of books near
the ceiling, I was able to gain access to Vaidyanathaswamy,
who was located near the top of the most inaccessible pile.
As you were probably recalling, he has a whole chapter on
the derived set operator on T_1 spaces. He weighs in on
the side of Narasimhan and Nievergelt, thus, on page 162:
"A set X is *discrete*, if D(X) = 0."
I guess he and Willard will just have to duke it out! :-)
(I'm not going to referee. In the unlikely event that I ever
have to write about this stuff, I suppose I will simply refer
to Vaidyanathaswamy and Narasimhan and Nievergelt for the def-
inition of the adjective "discrete", and warn the reader of
its ambiguity in an aside. That's normal enough practice in
a mathematical text, I think, and it will probably have to do.)
>Vaidyanathaswamy's book
>http://books.google.com/books?id=yDMipybQ64kC
(I bought three years ago, but have hardly glanced at it since.
I'll keep it handy from now on.)
>Scattered sets are sets that become emtpy after some countable
>iteration of the derived set operation (take intersection
>of all previous derived sets at the limit ordinal stages).
>
>I've posted a lot about these notions (isolated sets, the
>derived set operator, scattered sets, etc.), but the following
>posts seem to be the most relevant to what you're asking about.
>
>sci.math -- unique limit point in infinite bounded set
>(Dave L. Renfro; 13 & 17 June 2007)
>http://groups.google.com/group/sci.math/msg/23e6146526f92a98
>http://groups.google.com/group/sci.math/msg/1320c00bd89012ff
>
>sci.math -- Distrubutive Property of Limit Points
>(Dave L. Renfro; 10 October 2005)
>http://groups.google.com/group/sci.math/msg/3727727671de641e
>
>sci.math -- Question from Rudin
>(Dave L. Renfro; 15 May 2000)
>http://groups.google.com/group/sci.math/msg/a31647549c87f4cf
>
>sci.math -- MONOTONE SET FUNCTIONS, FIXED POINTS,
>AND CECH CLOSURE FUNCTIONS
>(Dave L. Renfro; 4 July 2005)
>http://groups.google.com/group/sci.math/msg/9eb76fc26a4cacff
>http://groups.google.com/group/sci.math/msg/550f93d6f4f252ed
References all perused, although I haven't the energy to work
through them yet. (Makes me wish the Internet had been around
when I was much younger, and could get sort of productively
obsessed with such things!) The last three in particular look
like I'll be referring to them, probably when I've learned
something about ordinal numbers. I'm vaguely aware of these
things (Cantor-Bendixson rank, etc.) from the introduction to
the Dover reprint of a couple of Cantor's papers.
--
Angus Rodgers
William Elliot
> First, the following terms all seem to be synonymous, in the sense
> that although different authors chose different terms from the list,
> no author also uses any of the other terms in a different sense:
>
> limit point
> accumulation point
> cluster point
>
> Is that right?
>
No. They are all different and different authors may use different
definitions. Hence is can be important to let your reader know what
definitions you or your text is using.
x is a limit point of a set A when x in cl A\x
x is a limit point of A iff for all open U nhood x,
A /\ U has more than one point, ie another point besides x.
x is an accumulation point of A when for all open U nhood x,
U /\ A is infinite.
When the space is T1, accumulation point and limit point are
equivalent. Thus for all who use metric spaces, including
analysts, there is no difference and they may also use or prefer
the term cluster point.
That however is a disservice because there's another definition of cluster
point, namely
x is a cluster point of a sequence { aj } when for all open U nhood x,
aj is frequently in the sequence, ie for all n in N, some m in N with
for all j > m, aj in U.
Finally there's the limit of a sequence { aj } which is much different
than limit point.
> Second, the set of all limit points (accumulation points, cluster
> points) of a subset A of a topological space X is always called
> the "derived set" of A. It is usually denoted by A'; however,
> G. F. Simmons, in /Introduction to Topology and Modern Analysis/,
> uses D(A) (having already used A' for the complement of A).
>
Notation will vary from author to author among all the sciences. There is
such is wide variance in usage between chemistry and physics for the same
or similar thermodynamics concepts, that a dictionary has been complied to
translate between the two sciences regarding thermodynamics.
> Third, a subset A of a topological space X is usually called
> "discrete" iff it forms a discrete space when given the subspace
> topology.
That is more common among the authors than the previous terms.
> However (and this brings me to the point at issue), Narasimhan
> and Nievergelt, in /Complex Analysis in One Variable/ (2nd ed.),
> Theorem 2 of section 1.3, define a subset Z of a connected open
> set D in C to be "discrete" iff it has only isolated points and
> is closed (as a subset of D, rather than C - although they don't
> say this explicitly).
>
That's how he's using the term, so note it and be sure when discussing
with others, to use the expression a discrete closed subspace.
> As I learned in the thread "Zeros of an analytic function", this
> property characterises those subsets Z of a domain D which are
> the zero sets of analytic functions D -> C.
>
That is why he has that specialized definition.
> In passing, let me check also that the terms "domain" and "region"
> are also synonymous, as both meaning an open, connected subset of
> C (which may also be required to be non-empty - but I won't worry
> too much about this optional extra condition). Is that right?
>
A domain is the set of elements for which a function is defined (unless
you're in a topic that uses partial functions). Region, not used in
topology, seems to be a term specific to analyst. The closest topology
term is "continuum", a compact connected Hausdorff space.
> I think the usage of Narasimhan and Nievergelt must be rejected
> (in spite of their authority), but what does this leave us with?
>
It's specific to their needs.
> Rudin, in /Real and Complex Analysis/ (3rd ed.), simply describes
> the characteristic property of zero sets of analytic functions
> D -> C as that of having no limit points in C. This is simple
> enough, I suppose, and better than my clumsier description of
> them, in that other thread, as "(i) discrete, and (ii) closed
> as a subset of D (but not necessarily as a subset of C)".
>
In topology A is a zero set of a space if it's the continuous pre-image of
0 of some real valued function. I would expect that in the specialized
environment of analyst, that the two are equivalent.
A subset S is zero set when some continuous f:S -> [0,1] (or R)
with A = f^-1(0).
> But is there not a simpler term - somewhere! - for a subset of a
> topological space whose derived set is empty? I can't find any.
>
A discrete open subspace?
> ("Type zero" was the only suggestion I came across - I forget
> where - and "scattered" also suggests itself, but seems to mean
> something not quite the same, although closely related.)
>
Let S be a space and define S^0 = S; S^(xi+1) = (S^(xi)');
S^(eta) = /\{ S^(xi) | xi < eta } when eta is a limit ordinal.
The scattering height of S,
sh S = min{ xi | S^(xi) = S^(xi + 1) }
sh S is always defined by use of Hartog's theorem.
S is scattered when S^(sh S) = nulset.
Angus Rodgers
<ma...@rdrop.remove.com> wrote:
>On Wed, 18 Feb 2009, Angus Rodgers wrote:
>
>> First, the following terms all seem to be synonymous, in the sense
>> that although different authors chose different terms from the list,
>> no author also uses any of the other terms in a different sense:
>>
>> limit point
>> accumulation point
>> cluster point
>>
>> Is that right?
>>
>No. They are all different and different authors may use different
>definitions. Hence is can be important to let your reader know what
>definitions you or your text is using.
>
>x is a limit point of a set A when x in cl A\x
>
>x is a limit point of A iff for all open U nhood x,
>A /\ U has more than one point, ie another point besides x.
>
>x is an accumulation point of A when for all open U nhood x,
>U /\ A is infinite.
>
>When the space is T1, accumulation point and limit point are
>equivalent. Thus for all who use metric spaces, including
>analysts, there is no difference and they may also use or prefer
>the term cluster point.
Excellent - that looks like a very useful distinction, worth
keeping in mind. However, I shall (when less tired!) have to
check all those topology texts again, because I am almost sure
that none of them makes the distinction you're making.
However, I wouldn't be surprised to find that Vaidyanathaswamy
does, because he gives particular attention to the role of the
T_1 axiom, for instance in proving that derived sets are closed.
No, Vaidyanathaswamy defines "accumulation point" just as you
have defined "limit point".
>That however is a disservice because there's another definition of cluster
>point, namely
>
>x is a cluster point of a sequence { aj } when for all open U nhood x,
>aj is frequently in the sequence, ie for all n in N, some m in N with
>for all j > m, aj in U.
That looks like another useful distinction, indeed; but again
I shall have to check actual usage, for my own peace of mind.
>[...]
>
>> Third, a subset A of a topological space X is usually called
>> "discrete" iff it forms a discrete space when given the subspace
>> topology.
>
>That is more common among the authors than the previous terms.
I haven't done even a preliminary check of how this term is used,
but that was my informal impression.
>> However (and this brings me to the point at issue), Narasimhan
>> and Nievergelt, in /Complex Analysis in One Variable/ (2nd ed.),
>> Theorem 2 of section 1.3, define a subset Z of a connected open
>> set D in C to be "discrete" iff it has only isolated points and
>> is closed (as a subset of D, rather than C - although they don't
>> say this explicitly).
>>
>That's how he's using the term, so note it and be sure when discussing
>with others, to use the expression a discrete closed subspace.
The trouble here is that this usage of "discrete" (implying closed)
is shared with Vaidyanathaswamy, so it seems I'll just have to get
used to "discrete" being more ambiguous than I thought.
>> As I learned in the thread "Zeros of an analytic function", this
>> property characterises those subsets Z of a domain D which are
>> the zero sets of analytic functions D -> C.
>>
>That is why he has that specialized definition.
Indeed. However, one would have thought that the property of having
an empty derived set (in any topology) would also have been of wider
interest, and acquired a name of its own.
>> In passing, let me check also that the terms "domain" and "region"
>> are also synonymous, as both meaning an open, connected subset of
>> C (which may also be required to be non-empty - but I won't worry
>> too much about this optional extra condition). Is that right?
>>
>A domain is the set of elements for which a function is defined (unless
>you're in a topic that uses partial functions).
I'll have to do some checking, but certainly Beardon uses the word
"domain" in this way - and (I must admit I hadn't noticed this) he
simply avoids its use in the (very well-established) general set-
theoretical sense. That's a pretty good reason for using "region"
instead, I think.
>Region, not used in
>topology, seems to be a term specific to analyst.
That seems OK, then. But I'll do some more checking (sigh) ...
>The closest topology
>term is "continuum", a compact connected Hausdorff space.
>
>> I think the usage of Narasimhan and Nievergelt must be rejected
>> (in spite of their authority), but what does this leave us with?
>>
>It's specific to their needs.
>
>> Rudin, in /Real and Complex Analysis/ (3rd ed.), simply describes
>> the characteristic property of zero sets of analytic functions
>> D -> C as that of having no limit points in C. This is simple
>> enough, I suppose, and better than my clumsier description of
>> them, in that other thread, as "(i) discrete, and (ii) closed
>> as a subset of D (but not necessarily as a subset of C)".
>>
>In topology A is a zero set of a space if it's the continuous pre-image of
>0 of some real valued function. I would expect that in the specialized
>environment of analyst, that the two are equivalent.
>
>A subset S is zero set when some continuous f:S -> [0,1] (or R)
>with A = f^-1(0).
I just made up the term "zero set" (as it was awkward to keep using
"set of zeros"). I should probably have explained this, especially
as I was posting to try to clarify terminology.
>> But is there not a simpler term - somewhere! - for a subset of a
>> topological space whose derived set is empty? I can't find any.
>>
>A discrete open subspace?
That doesn't seem clear at all. First, I don't see what "open"
has to do with it. (Sorry if I'm being thick, but I'm rather tired
and stressed!) Second, the property in question, unlike the property
of being "discrete" (in what I thought was its usual sense), is not
a property of the subspace topology of the set in question, but of
the set in question in relation to the topology of its containing
space. Sorry, that was clumsy! I mean, whether a subset A of a
space X has or does not have limit points in X is not discernible
purely in terms of the topology of A inherited from that of X as
one of its subspaces. (Still clumsy, but I'm exhausted!)
>> ("Type zero" was the only suggestion I came across - I forget
>> where - and "scattered" also suggests itself, but seems to mean
>> something not quite the same, although closely related.)
>>
>Let S be a space and define S^0 = S; S^(xi+1) = (S^(xi)');
>S^(eta) = /\{ S^(xi) | xi < eta } when eta is a limit ordinal.
>
>The scattering height of S,
>sh S = min{ xi | S^(xi) = S^(xi + 1) }
>
>sh S is always defined by use of Hartog's theorem.
>
>S is scattered when S^(sh S) = nulset.
Yes, I saw this definition somewhere, but I think I also saw some
author using "scattered" in a different and simpler sense. (If so,
however, it was idiosyncratic of him, and so it can pretty much be
ignored. I don't think I'll search for it again!)
--
Angus Rodgers
William Elliot
> <ma...@rdrop.remove.com> wrote:
>> On Wed, 18 Feb 2009, Angus Rodgers wrote:
>>
>> x is a limit point of a set A when x in cl A\x
>>
>> x is a limit point of A iff for all open U nhood x,
>> A /\ U has more than one point, ie another point besides x.
>>
>> x is an accumulation point of A when for all open U nhood x,
>> U /\ A is infinite.
>>
>> When the space is T1, accumulation point and limit point are
>> equivalent. Thus for all who use metric spaces, including
>> analysts, there is no difference and they may also use or prefer
>> the term cluster point.
>
> Excellent - that looks like a very useful distinction, worth
> keeping in mind. However, I shall (when less tired!) have to
> check all those topology texts again, because I am almost sure
> that none of them makes the distinction you're making.
>
> However, I wouldn't be surprised to find that Vaidyanathaswamy
> does, because he gives particular attention to the role of the
> T_1 axiom, for instance in proving that derived sets are closed.
>
> No, Vaidyanathaswamy defines "accumulation point" just as you
> have defined "limit point".
>
accumulation point for the other. You will never get an universal
agreement on this. The best policy is to clearly define your usage of
these types of terms whenever you've a new reader. For other definition,
it's helpful to be over expressive. For example, normal T1.
>> That however is a disservice because there's another definition of cluster
>> point, namely
>>
>> x is a cluster point of a sequence { aj } when for all open U nhood x,
>> aj is frequently in the sequence, ie for all n in N, some m in N with
>> for all j > m, aj in U.
>
> That looks like another useful distinction, indeed; but again
> I shall have to check actual usage, for my own peace of mind.
>
Again usage will vary. Some use cluster point for limit or accumulation
points.
>>> However (and this brings me to the point at issue), Narasimhan
>>> and Nievergelt, in /Complex Analysis in One Variable/ (2nd ed.),
>>> Theorem 2 of section 1.3, define a subset Z of a connected open
>>> set D in C to be "discrete" iff it has only isolated points and
>>> is closed (as a subset of D, rather than C - although they don't
>>> say this explicitly).
>>>
>> That's how he's using the term, so note it and be sure when discussing
>> with others, to use the expression a discrete closed subspace.
>
> The trouble here is that this usage of "discrete" (implying closed)
> is shared with Vaidyanathaswamy, so it seems I'll just have to get
> used to "discrete" being more ambiguous than I thought.
>
Always take care to note the author's use of words before reading from the
text, especially if you're reading just sections.
>>> As I learned in the thread "Zeros of an analytic function", this
>>> property characterises those subsets Z of a domain D which are
>>> the zero sets of analytic functions D -> C.
>>>
>> That is why he has that specialized definition.
>
> Indeed. However, one would have thought that the property of having
> an empty derived set (in any topology) would also have been of wider
> interest, and acquired a name of its own.
>
The property of having an empty derived set is relative to the space the
set is embedded into. For example A = { 1/n | n in N } as subset of
[0,oo) has A' = {0} and as subset of (0,oo) has A' = nulset.
>>> In passing, let me check also that the terms "domain" and "region"
>>> are also synonymous, as both meaning an open, connected subset of
>>> C (which may also be required to be non-empty - but I won't worry
>>> too much about this optional extra condition). Is that right?
>>>
>> A domain is the set of elements for which a function is defined (unless
>> you're in a topic that uses partial functions).
>
> I'll have to do some checking, but certainly Beardon uses the word
> "domain" in this way - and (I must admit I hadn't noticed this) he
> simply avoids its use in the (very well-established) general set-
> theoretical sense. That's a pretty good reason for using "region"
> instead, I think.
>
As analysis is different than topology, you are to expect some divergence
of terminology.
>> Region, not used in
>> topology, seems to be a term specific to analyst.
>
> That seems OK, then. But I'll do some more checking (sigh) ...
>
>>> Rudin, in /Real and Complex Analysis/ (3rd ed.), simply describes
>>> the characteristic property of zero sets of analytic functions
>>> D -> C as that of having no limit points in C. This is simple
>>> enough, I suppose, and better than my clumsier description of
>>> them, in that other thread, as "(i) discrete, and (ii) closed
>>> as a subset of D (but not necessarily as a subset of C)".
>>>
>> In topology A is a zero set of a space if it's the continuous pre-image of
>> 0 of some real valued function. I would expect that in the specialized
>> environment of analyst, that the two are equivalent.
>>
>> A subset S is zero set when some continuous f:S -> [0,1] (or R)
>> with A = f^-1(0).
>
> I just made up the term "zero set" (as it was awkward to keep using
> "set of zeros"). I should probably have explained this, especially
> as I was posting to try to clarify terminology.
>
Sorry, you've been preempted.
>>> But is there not a simpler term - somewhere! - for a subset of a
>>> topological space whose derived set is empty? I can't find any.
>>>
>> A discrete open subspace?
>
> That doesn't seem clear at all.
It's worse than that. It's not correct that a discrete open subspace has
an empty derived set when viewed as a set within the larger space.
> First, I don't see what "open" has to do with it. Second, the property
> in question, unlike the property of being "discrete" (in what I thought
> was its usual sense), is not a property of the subspace topology of the
> set in question, but of the set in question in relation to the topology
> of its containing space. Sorry, that was clumsy! I mean, whether a
> subset A of a space X has or does not have limit points in X is not
> discernible purely in terms of the topology of A inherited from that of
> X as one of its subspaces. (Still clumsy, but I'm exhausted!)
>
That is correct. A quick grab from my notes yields some facts.
If the space is compact, and A' = nulset, then A is finite.
Conversely, if space is T1 and A is finite, then A' = nulset.
Take it easy, use the expression empty derived set.
A is nowhere dense when int cl A = nulset. There is no expression
for when int A = nulset other than empty interior set.
Analysts especially use the expression compact to mean compact closed.
Topologists are not so wanton. There is theorem that compact subset of a
Hausdorff space is closed. Thus to analysts, whose spaces are all
Hausdorff, compact sets are closed.
Angus Rodgers
<ma...@rdrop.remove.com> wrote:
>The property of having an empty derived set is relative to the space the
>set is embedded into. For example A = { 1/n | n in N } as subset of
>[0,oo) has A' = {0} and as subset of (0,oo) has A' = nulset.
We're agreed on that.
>>>> In passing, let me check also that the terms "domain" and "region"
>>>> are also synonymous, as both meaning an open, connected subset of
>>>> C (which may also be required to be non-empty - but I won't worry
>>>> too much about this optional extra condition). Is that right?
>>>>
>>> A domain is the set of elements for which a function is defined (unless
>>> you're in a topic that uses partial functions).
>>
>> I'll have to do some checking, but certainly Beardon uses the word
>> "domain" in this way - and (I must admit I hadn't noticed this) he
>> simply avoids its use in the (very well-established) general set-
>> theoretical sense. That's a pretty good reason for using "region"
>> instead, I think.
>>
>As analysis is different than topology, you are to expect some divergence
>of terminology.
Indeed, but set theory is so universal, in analysis as elsewhere,
that to give one of its most-used terms a double meaning like this
seems most unwise.
I'm surprised it never struck me until you pointed it out!
I think this illustrates how I tend to become virtually hypnotised
by the author of whatever textbook I'm reading at the moment: this
is one reason why I need to have a lot of books around me to widen
my perspective.
>>>> Rudin, in /Real and Complex Analysis/ (3rd ed.), simply describes
>>>> the characteristic property of zero sets of analytic functions
>>>> D -> C as that of having no limit points in C. This is simple
>>>> enough, I suppose, and better than my clumsier description of
>>>> them, in that other thread, as "(i) discrete, and (ii) closed
>>>> as a subset of D (but not necessarily as a subset of C)".
>>>>
>>> In topology A is a zero set of a space if it's the continuous pre-image of
>>> 0 of some real valued function. I would expect that in the specialized
>>> environment of analyst, that the two are equivalent.
>>>
>>> A subset S is zero set when some continuous f:S -> [0,1] (or R)
>>> with A = f^-1(0).
>>
>> I just made up the term "zero set" (as it was awkward to keep using
>> "set of zeros"). I should probably have explained this, especially
>> as I was posting to try to clarify terminology.
>>
>Sorry, you've been preempted.
Fine by me! It was just a temporary improvisation, and one I
now slightly regret.
>Take it easy, use the expression empty derived set.
>
>A is nowhere dense when int cl A = nulset. There is no expression
>for when int A = nulset other than empty interior set.
Good point.
--
Angus Rodgers
Dave L. Renfro
>> Vaidyanathaswamy's book
>> http://books.google.com/books?id=yDMipybQ64kC
Angus Rodgers wrote (in part):
> (I bought three years ago, but have hardly glanced
> at it since. I'll keep it handy from now on.)
Vaidyanathaswamy's book is filled with a lot of
interesting but little known (in current times,
at least) material, but it's probably a bit too
far down the topology tree for me to recommend
you spending much time with it, at least if you're
currently pursuing complex analysis. On the other
hand, it's perfect for someone like William Elliot,
and if he doesn't have a copy of the book, I strongly
encourage him to get a copy. Dover began reprinting
it a few years ago, so it's both easy to get and
not very expensive. I've known about his book since
probably the 1970s, but it was only around 1990 or
so that I made a careful study of parts of it,
in particular the chapter on the derived set
operator. At one time I thought I might be able
to do something for a Ph.D. based on the material
he includes (and some more similar stuff in a two
or three of his published papers on "local function
properties"), but nothing of significance suggested
itself and the more I dug into the subject, the
more I learned that others have mined it very
deeply. Still, I have a 60+ page (LaTeX typed)
beginnings of a comprehensive survey of this sort
of stuff (e.g. see the post below).
sci.math -- Set of bilateral condensation points of
uncountable subsets of R
(Dave L. Renfro; 27 March 2007)
http://groups.google.com/group/sci.math/msg/adad4f80670bd5d0
While I'm at it, I'd like to make a plug for a book
I bought a few months ago but have only recently
gotten around to look at carefully:
Charles Chapman Pugh, "Real Mathematical Analysis",
Undergraduate Texts in Mathematics, Springer-Verlag,
2002, xi + 437 pages.
This book is VERY well written, in the same way
that Stephen Abbott's 2001 "Understanding Analysis"
book is. [Back in 2003 I called Abbott's book "the best
written introductory analysis book that's appeared
in the past couple of decades".]
alt.math.undergrad -- epsilon-delta book
(Dave L. Renfro; 5 January 2003)
http://groups.google.com/group/alt.math.undergrad/msg/928de25882e28e92
Pugh is pitched at a little higher level than Abbott's book.
Pugh contains a large number of very interesting exercises,
a far greater than average number of diagrams (and the
diagrams are quite useful and well done), and a lot of
"here's what is really going on" type of comments directed
to the reader (who is assumed to be a person and not a
machine).
Dave L. Renfro
Dave L. Renfro
> Vaidyanathaswamy's book is filled with a lot of
> interesting but little known (in current times,
> at least) material, but it's probably a bit too
> far down the topology tree for me to recommend
> you spending much time with it, at least if you're
> currently pursuing complex analysis.
By "too far down the topology tree", I don't mean
it's too topology oriented (it's a book on topology,
after all), but that it's focus is mostly on the
set-theoretical aspects of the fundamental aspects
of general topology -- boolean algebras, sigma-fields,
partial orders, lattices, and such. While there is much
in the later parts of the book (where function spaces
are studied) that would be useful, most of this material
is probably (for an "outsider") better studied in more
recently written texts (Willard, Munkres, etc.). There
is no index either, only a list of 25 or 30 names, so
it's very hard to look anything up in the book unless
you're fairly familar with it.
Dave L. Renfro
Angus Rodgers
<renf...@cmich.edu> wrote:
>While I'm at it, I'd like to make a plug for a book
>I bought a few months ago but have only recently
>gotten around to look at carefully:
>
>Charles Chapman Pugh, "Real Mathematical Analysis",
>Undergraduate Texts in Mathematics, Springer-Verlag,
>2002, xi + 437 pages.
>
>This book is VERY well written, in the same way
>that Stephen Abbott's 2001 "Understanding Analysis"
>book is. [Back in 2003 I called Abbott's book "the best
>written introductory analysis book that's appeared
>in the past couple of decades".]
>
>alt.math.undergrad -- epsilon-delta book
>(Dave L. Renfro; 5 January 2003)
>http://groups.google.com/group/alt.math.undergrad/msg/928de25882e28e92
>
>Pugh is pitched at a little higher level than Abbott's book.
>Pugh contains a large number of very interesting exercises,
>a far greater than average number of diagrams (and the
>diagrams are quite useful and well done), and a lot of
>"here's what is really going on" type of comments directed
>to the reader (who is assumed to be a person and not a
>machine).
It was one of the books I thought of getting in the 2006 Yellow
Sale (my annual orgy!), but I had to be very selective (I'm on
a strict budget), so missed it. Perhaps it'll come round again.
From the first few pages at Amazon, I get the impression that he's
exasperated with calculus teaching, he perhaps means "real" in the
title of the book in two senses, and is, a little bit (like Hardy,
according to Littlewood) like "a missionary talking to cannibals".
Am I right? The first few pages may give a misleading impression,
or perhaps my famed reading comprehension is playing up again! ;-)
--
Angus Rodgers
Dave L. Renfro
> From the first few pages at Amazon, I get the impression
> that he's exasperated with calculus teaching, he perhaps
> means "real" in the title of the book in two senses, and is,
> a little bit (like Hardy, according to Littlewood) like
> "a missionary talking to cannibals". Am I right? The first
> few pages may give a misleading impression, or perhaps my
> famed reading comprehension is playing up again! ;-)
I'll get back to you after I have a chance to look at
the book again. Maybe I'll just bring it to work with
me tomorrow. Right now I'm at work to take care of a
few things (early Sunday A.M.) because we're on a tight
deadline schedule for something due Tuesday and I have
other things already scheduled later today (a couple
of hours at the gym, some tutoring appointments, etc.),
plus by taking care of some things now I can hand it off
to someone else (who'll be in later today) who can't do
what they need to do until I spend a few hours taking
care of my part this morning.
Dave L. Renfro
Dave L. Renfro
> From the first few pages at Amazon, I get the impression
> that he's exasperated with calculus teaching, he perhaps
> means "real" in the title of the book in two senses, and is,
> a little bit (like Hardy, according to Littlewood) like
> "a missionary talking to cannibals". Am I right? The first
> few pages may give a misleading impression, or perhaps my
> famed reading comprehension is playing up again! ;-)
I've looked through the book [Pugh's "Real Mathematical
Analysis"] a little (about 5 minutes) just now and I don't
see this view coming through, so if he did say something
along these lines, it's probably very little and in one
place. Here's an example of one of the exercises, this
one being a bit silly and non-traditional, but it'll
serve to illustrate my point about many of his exercises
being fairly unique and interesting:
Exercise 130 on p. 135: "Write jingles at least as good
as the following."
"When a set
in the plane
is closed and bounded
you can always draw
a curve around it."
"If a clopen set can be detected
Your metric space is disconnected."
[I've omitted two longer jingles he also included.]
Here's another one:
Exercise 21(b) on p. 254: "Given constants a, b > 0
define f_{a,b}(x) = (x^a)*sin(x^b) for all x > 0.
For which sets of (a,b) in (0,oo)^2 is the family
equicontinuous?"
And another one:
Exercise 35 on p. 256: "(a) Prove Borel's Lemma:
given any sequence whatsoever of real numbers
(a_r) there is a smooth [i.e. C^infinity] function
f:R --> R such that f^(r)(0) = a_r [r'th derivative
at 0 is equal to a_r]. (b) Infer that there are many
Taylor series with radius of convergence R = 0.
(c) Construct a smooth function whose Taylor series
at every x has radius of convergence R = 0."
Dave L. Renfro
Angus Rodgers
<renf...@cmich.edu> wrote:
>Angus Rodgers wrote (in part):
>
>> [...]
>
>I've looked through the book [Pugh's "Real Mathematical
>Analysis"] a little (about 5 minutes) just now and I don't
>see this view coming through, so if he did say something
>along these lines, it's probably very little and in one
>place.
(That place could well be only in my mind.) :-)
>Here's an example of one of the exercises, this
>one being a bit silly
Oh, good! I'm in no mood for actual maths at the moment.
>and non-traditional, but it'll
>serve to illustrate my point about many of his exercises
>being fairly unique and interesting:
>
>Exercise 130 on p. 135: "Write jingles at least as good
>as the following."
>
>"When a set
>in the plane
>is closed and bounded
>you can always draw
>a curve around it."
>
>"If a clopen set can be detected
>Your metric space is disconnected."
>
>[I've omitted two longer jingles he also included.]
However much it twists about
A Jordan curve marks 'in' from 'out'.
The circle's image winds around
Just once for every zero found.
You can't achieve an upper bound
When there's an open disc around!
"The integral is always zero."
That's Cauchy's theorem - he's my hero!
(Not getting zero, as expected?
How simply's your domain connected?)
(Talking of "heroes", I have to watch it now, so you're spared
any more of these.)
--
Angus Rodgers
Angus Rodgers
<renf...@cmich.edu> wrote:
>Exercise 21(b) on p. 254: "Given constants a, b > 0
>define f_{a,b}(x) = (x^a)*sin(x^b) for all x > 0.
>For which sets of (a,b) in (0,oo)^2 is the family
>equicontinuous?"
I just looked up the definition of equicontinuity in Rudin (it was
what I imagined it might be), and as far as I can see, the answer
is: any subset of {(a, 1 - a): 0 < a < 1} (unless equicontinuity
presupposes that the set is nonempty - but Rudin doesn't say so).
If there is a non-empty solution, then every f = f_{a,b} in it must
be uniformly continuous on (0,oo). Since f is differentiable, f'
must be bounded on (0,oo). But:
f'(x) = a*x^{a - 1}*sin(x^b) + b*x^{a + b - 1}*cos(x^b)
If n is a positive integer, and x = (2*n*pi)^{1/b}, then f'(x) =
b*x^{a + b - 1}, which (because b > 0, and x can be arbitrarily
large) is arbitrarily large, unless a + b - 1 <= 0.
So, suppose a + b - 1 <= 0. As x -> 0, x^b -> 0, sin(x^b) =
x^b + O(x^{3b}), and cos(x^b) = 1 + O(x^{2b}), therefore:
f'(x) = (a + b)*x^{a + b - 1} + O(x^{a + 3b - 1})
~ (a + b)*x^{a + b - 1}
Since a + b > 0, this is only unbounded if a + b = 1.
On the other hand, if a + b = 1, then, for all x > 0:
|f'(x)| <= a*|sin(x^{1 - a})|/x^{1 - a} + (1 - a)*|cos(x^{1 - a})|
< 1 (because |sin(u)| < u for all u > 0)
and so |f(x) - f(y)| <= |x - y| for all x, y > 0.
(Hope I haven't done anything really silly, as I'm in an even more
distracted frame of mind than usual. I'm going to get back to my
"cranky" stuff now! I'll try to have a look at the other exercise
later.)
--
Angus Rodgers
Angus Rodgers
<renf...@cmich.edu> wrote:
>Exercise 35 on p. 256: "(a) Prove Borel's Lemma:
>given any sequence whatsoever of real numbers
>(a_r) there is a smooth [i.e. C^infinity] function
>f:R --> R such that f^(r)(0) = a_r [r'th derivative
>at 0 is equal to a_r].
Astonishing result, which I'd never heard of. (Not too surprising,
as I've never completed a course of study in real analysis. Wish
I had now!) Torture trying to prove it. I haven't had a single
usable idea. (Also not surprising, as today has been an off-day.)
I did, however, come across a reference (in DePree & Swartz):
@article{meyerson:borel,
author = "M. D. Meyerson",
title = "Every Power Series is a {T}aylor Series",
journal = "Amer. Math. Monthly",
volume = "88",
number = "1",
year = "1981",
pages = "51--52"}
But I haven't got JSTOR access, so I can't cheat. (Damn!) I may
perhaps ask for a hint, if I still can't make any progress even
after sleeping on it. But I want to give the old unconscious a
chance first.
--
Angus Rodgers
Dave L. Renfro
>> Exercise 35 on p. 256: "(a) Prove Borel's Lemma:
>> given any sequence whatsoever of real numbers
>> (a_r) there is a smooth [i.e. C^infinity] function
>> f:R --> R such that f^(r)(0) = a_r [r'th derivative
>> at 0 is equal to a_r].
Angus Rodgers wrote (in part):
> Astonishing result, which I'd never heard of. (Not too surprising,
> as I've never completed a course of study in real analysis. Wish
> I had now!) Torture trying to prove it. I haven't had a single
> usable idea. (Also not surprising, as today has been an off-day.)
> I did, however, come across a reference (in DePree & Swartz):
Pugh's book proves a standard result about the rate of growth
of the taylor coefficients at a point (as a function of n,
where n is the degree of the term) in order for the function
to be analytic or non-analytic at a point, and for the problem
at hand Pugh gives a hint, which I don't know the details of
(I don't have the book with me now) but I think it's something
like a condensation of singularities construction (google the
phrase in google-groups -- I've briefly discussed and given
references to this method several times) that amounts to getting
the coefficients to grow too quickly at each rational number.
Because "analytic at a x=b" is equivalent to "analytic in
some neighborhood of x=b", getting non-analyticity at each
point in a set dense in the reals will do it.
About 7 years ago I posted a length essay (in 2 parts) on
C-infinity functions that are nowhere analytic in sci.math.
However, the posts were made at Math Forum and, for some
reason (as was also the case with many posts of mine from
around 2001 to 2003), the posts never made it to the google
sci.math archive. However, I notice that just a few days
before I posted these essays, I announced I'd be doing so
and my announcement _is_ in google's sci.math archive:
http://groups.google.com/group/sci.math/msg/508e47215251c413
Anyway, here are the essays. For some reason certain accent
symbols on non-English names/titles are all messed up, although
when the posts originally appeared I'm certain they came through
at Math Forum's archive pages fine.
ESSAY ON NOWHERE ANALYTIC C-INFINITY FUNCTIONS (part 1)
http://mathforum.org/kb/message.jspa?messageID=387148
ESSAY ON NOWHERE ANALYTIC C-INFINITY FUNCTIONS (part 2)
http://mathforum.org/kb/message.jspa?messageID=387149
Below are the contents, followed by some excerpts from
part 1.
CONTENTS (for part 1)
I. TWO WAYS THAT A C-INFINITY FUNCTION CAN FAIL TO BE ANALYTIC
II. EARLY HISTORY OF NOWHERE ANALYTIC C-INFINITY FUNCTIONS
III. ZAHORSKI'S 1947 CHARACTERIZATION
IV. MOST C-INFINITY FUNCTIONS ARE NOWHERE ANALYTIC
V. UNEXPLORED AREAS AND OTHER RESULTS
VI. REFERENCES (1-35)
CONTENTS (for part 2)
I. REFINEMENTS ON THE TWO WAYS THAT A C-INFINITY FUNCTION
CAN FAIL TO BE ANALYTIC
II. GENERALIZATIONS OF ZAHORSKI'S 1947 CHARACTERIZATION
III. ZERO SETS OF C-INFINITY FUNCTIONS AND ULAM'S PROBLEM
IV. MORE CONCERNING "MOST C-INFINITY FUNCTIONS ARE NOWHERE
ANALYTIC"
V. REFERENCES (36-47)
SOME EXCERPTS FROM part 1:
Every C-infinity function has a formal Taylor series expansion
about each point. If this Taylor expansion converges to the
original function in neighborhood of x=b, then the function is
said to be analytic at x=b. There are two ways this can fail:
(C) The Taylor expansion converges in a neighborhood
of x=b, but in no neighborhood of x=b does the
Taylor expansion converge to the given function.
(P) The Taylor expansion fails to converge in every
neighborhood of x=b (i.e. the Taylor series at
x=b has a zero radius of convergence).
Following Zahorski [35], we classify the non-analytic points
of a C-infinity function into the following two categories.
A point of non-analyticity is called a (C)-point (for Cauchy)
if it belongs to (C) above. A point of non-analyticity is called
a (P)-point (for Pringsheim) if it belongs to (P) above.
*********************** [snip] *******************************
III. ZAHORSKI'S 1947 CHARACTERIZATION
None of these early examples was shown to have a (C)-point
everywhere or to have a (P)-point everywhere. Boas proved
in 1935 (see [4] or p. 192 of [5]) that the (P)-points
of a nowhere analytic C-infinity function are dense in R.
Therefore, it is not possible to have an example in which every
point is a (C)-point. However, the possibility that every point
could be a (P)-point remained open. The first example of a
C-infinity function such that every point is a (P)-point was
given by Cartan in 1940 ([6], pp. 20-22). In 1947 Zahorski [35]
gave the following characterization for the non-analytic points
of a C-infinity function. [An announcement [34] of this result was
made in 1946.]
THEOREM (Zahorski): A necessary and sufficient condition for
two sets of real numbers C and P to be the (C)-points
and the (P)-points, respectively, of some C-infinity
function is that the following four properties hold:
(a) C is a first category F_sigma set.
(b) P is a G_delta set.
(c) C is disjoint from P.
(d) C union P is closed in R.
[[ Zahorski died on May 8, 1998 at the age of 84. I
believe his work on the singularities of C-infinity
functions was part of his Ph.D. research. ]]
We mention three corollaries.
** If P is empty in some interval, then the set of (C)-points in
that interval is a relatively closed first category subset of
that interval. Since every closed first category set is nowhere
dense, no such function can be nowhere analytic in that interval.
Hence, if a function is nowhere analytic and C-infinity, then
every interval must contain some (P)-points, and we get the
result that Boas proved in 1935.
** Another corollary is the existence of a C-infinity function
having a (P)-point everywhere: Choose C to be empty and P = R.
** Still another corollary is that there exist C-infinity functions
belonging to each of the following categories (recall Pringsheim's
examples in Section II):
(i) C is c-dense in R and P is empty.
(ii) C is empty and P is c-dense in R.
(iii) Both C and P are c-dense in R.
For multivariable versions of Zahorski's theorem, see Bartczak [1],
Schmets/Valdivia [27] [28], Siciak [29], and H. Zahorska [33].
IV. MOST C-INFINITY FUNCTIONS ARE NOWHERE ANALYTIC
The Baire category theorem can be used to prove the existence of
nowhere analytic C-infinity functions, and this has been
re-discovered several times. Using the standard metric on
C-infinity (more precisely, any metric that generates the topology
of uniform convergence for all orders of derivatives on compact
sets), Morgenstern [22] (1954) gave a concise proof that the
Baire-typical C-infinity function is nowhere analytic. An outline
of a proof can be found on pp. 301-302 of Dugundji [12], with
Morgenstern's name mentioned. However, Dungundji states that
the particular proof he presents is due to Salzmann and Zeller,
apparently from a personal communication. For an expanded version
in English of Morgenstern's original proof, see pp. 95-97 of
Jones [16].
Next up, we have Christensen [9] (1972), who proves the same
result, unaware of Morgenstern. Then we have Darst [10] (1973),
who was apparently unaware of both Morgenstern's and Christensen's
papers. However, Darst followed this up with [11] (1974), where a far
stronger Baire-typical result is proved. [See page 26. The result
I am alluding to only shows up in the proof of a certain theorem,
not in any of his theorem statements.] Darst shows that certain
quasi-analytic classes have a Baire-typical set of functions that
are nowhere quasi-analytic relative to other quasi-analytic notions.
Next, we have Cater [7] (1984), who was aware of both Christensen's
and Darst's papers, and probably also of Morgenstern's paper.
[Darst writes (p. 618): "Two references in English are ...".]
Cater's paper is well written and his proof has all the details
worked out. Siciak [29] (1986), who was aware of Morgenstern's
paper, proves a Baire-typical multi-dimensional analog of
Morgenstern's result (theorem 10 on p. 144). After this, there
is Bernal [2] (1987). Bernal states that a corollary of the main
result in his paper is that the Baire-typical C-infinity function
is nowhere analytic. [Bernal cites Cater, Christensen, and Darst
in his bibliography.] Finally, Ramsamujh [23] (1991) proves that
the Baire-typical C-infinity function is nowhere analytic, unaware
at that time of any of the preceding papers.
Aside from simply being nowhere analytic, can we say anything about
the (C)-points and the (P)-points of the Baire-typical C-infinity
function? As far as I can tell, it appears that all the sources I
mentioned in the previous two paragraphs, except for Bernal [2],
Ramsamujh [23], and Siciak [29], prove only that the Baire-typical
C-infinity function has a dense set of (P)-points. From Zahorski's
theorem we know that the set of (P)-points is a G_delta set
(actually, this is immediate from the definition), and so we have
the interesting observation that the Baire-typical C-infinity
function has a Baire-typical set of (P)-points. In fact,
Christensen [9] explicitly states his result in this way.
However, Bernal [2], Ramsamujh [23], and Siciak [29] manage to
prove more. They actually prove that EVERY point is a (P)-point
for the Baire-typical C-infinity function. Thus, the Baire-typical
C-infinity function has no (C)-points at all. Although this
completely settles the matter, the result is unfortunate because
it closes the door on the possibility of investigating the Lebesgue
measure (or Hausdorff dimension, if measure zero) of the sets of
(C)-points and (P)-points of the Baire-typical C-infinity function.
Dave L. Renfro
Dave L. Renfro
>> Exercise 35 on p. 256: "(a) Prove Borel's Lemma:
>> given any sequence whatsoever of real numbers
>> (a_r) there is a smooth [i.e. C^infinity] function
>> f:R --> R such that f^(r)(0) = a_r [r'th derivative
>> at 0 is equal to a_r]. (b) Infer that there are many
>> Taylor series with radius of convergence R = 0.
>> (c) Construct a smooth function whose Taylor series
>> at every x has radius of convergence R = 0."
Dave L. Renfro later wrote (in part):
> Pugh's book proves a standard result about the rate of growth
> of the taylor coefficients at a point (as a function of n,
> where n is the degree of the term) in order for the function
> to be analytic or non-analytic at a point, and for the problem
> at hand Pugh gives a hint, which I don't know the details of
> (I don't have the book with me now) but I think it's something
> like a condensation of singularities construction (google the
> phrase in google-groups -- I've briefly discussed and given
> references to this method several times) that amounts to getting
> the coefficients to grow too quickly at each rational number.
> Because "analytic at a x=b" is equivalent to "analytic in
> some neighborhood of x=b", getting non-analyticity at each
> point in a set dense in the reals will do it.
I brought Pugh's book with me today . . .
Hint given for (a): "Try f = SUM [(beta_k)(x) * a_k * x^k / k!]
where beta_k is a well chosen bump function."
Hint given for (c): "Try SUM [(beta_k)(x) * e(x + q_k)]
where {q_1, q_2, ...} = rationals."
The function 'e' is defined back on p. 212 (I had to
look around a bit before I found it):
e(x) = exp(-1/x) if x > 0
= 0 if x <= 0
Note that e(x) is a C^infinity function that is
not analytic at x = 0. [Both parts of this can be
proved similar to the standard proof of the same results
for exp(-1/x^2), defined to be 0 at x = 0 -- use induction
to show that the n'th derivative of e(x) for x > 0 is a
rational function times e(x), and then things fall out
fairly easily.]
Dave L. Renfro