What is compact set?
Linda
In topology, compact set is an important concept. And in text book, it
is defined as if every open cover of set E has a finited open subcover,
then E is said to be a compact set. But what this really mean? I can
not understand this definition very well. For example, if you ask me
what a set is, I can imagine what it is like and explain very well; but
if you ask me what a compact set is, I can imagine nothing but recite
its definition awkwardly.
Can any one help to explain this concept and explain where is is used?
Thank in advance.
Robert Low
Linda wrote:
> In topology, compact set is an important concept. And in text book, it
> Can any one help to explain this concept and explain where is is used?
> Thank in advance.
You might find
http://en.wikipedia.org/wiki/Compact
informative.
Dave L. Renfro
As for where it's used, think about all the results
in elementary and advanced calculus in which the
hypothesis includes a closed and bounded interval
(real line) or a closed and bounded set (any R^n).
* A continuous real-valued function on a closed
and bounded set has both a maximum value and
a minimum value. In particular, any such function
is bounded, a fact that is often useful in proofs.
[Let M be such that |f| <= M on the set and
epsilon > 0. Choose delta = epsilon/M. Then ...]
* A continuous function on a closed and bounded
set is uniformly continuous on that set.
One way to understand this definition is due to
Hermann Weyl: "If a city is compact, it can be guarded
by a finite number of arbitrarily near-sighted policemen."
The essential idea about compactness is that it allows
you, under certain circumstances (usually in proofs),
to reduce situations involving some type of infinite
aspect of a problem to some corresponding finite
aspect, which can then be handled by a wider variety
of tools. Stephen Semmes manuscript "Where the Buffalo
Roam: Infinite Processes and Infinite Complexity"
discusses this view a little, although he also deals
with many other examples where one can make the
reduction from an infinite case to a finite one.
http://front.math.ucdavis.edu/math.CA/0302308
An equivalent formulation of compactness (use De Morgan's
laws) is that every nonempty collection of closed sets
with the finite intersection property (this means that
every nonempty finite subcollection has a nonempty
intersection) has a nonempty intersection.
There are many equivalent formulations of compactness
in R^n (many of which are also equivalent in metric
spaces, but not equivalent in topological spaces)
that might help you with the idea. These are easy
to google up or find in texts on advanced calculus,
functional analysis, metric spaces, topology. Some
examples are limit point compact, countably compact,
Lindelof compact, sequentially compact, Bolzano-Weierstrass
compact, etc.
A nice historical survey of compactness is:
Manya Raman, "Understanding Compactness: A Historical
Perspective", Masters Thesis (UC-Berkeley, 1997), 34 pages.
http://socrates.berkeley.edu/~manya/compact/
However, this URL no longer works for some reason, and
I wasn't able to find an alternate location for this
Thesis on the internet. [I'm glad I made a print copy
of this years ago. Unfortunately, I didn't keep the
.ps file for the paper. Given how transitory things
are on the internet, I think I'm going to start saving
the digital files of things I come across that for me
are worth making a print copy of.]
Dave L. Renfro
JEMebius
Linda wrote:
Mr Robert Low already mentioned the Wikipedia article at
http://en.wikipedia.org/wiki/Compact .
The concept of "compact" is in mathematics the most important bridge
between the finite and the infinite.
Wikipedia speaks of "compact" as the next best to "finite".
The word stems from the Latin "compactus", which means "thick-set". The
Latin word "compactus" stems from "compingo" = "I push together". This
meaning explains IMO the best how the term "compact" came into use in
mathematics: think of the classical situation of a closed segment S =
[a, b] of the real line. An infinite subset T of S has necessarily at
least one condensation point in S, a point where points of T were pushed
together, so to speak (Bolzano-Weierstrass theorem in 1D).
Now it is time for you to analyze thoroughly a proof of the
Bolzano-Weierstrass theorem
(see http://en.wikipedia.org/wiki/Bolzano-Weierstrass_theorem)
and to pinpoint the place of the finite-subcover property in that proof.
Happy studies: Johan E. Mebius
Stephen J. Herschkorn
From Munkres:
The notion of compactness is not nearly so natural as that of
connectedness. From the beginnings of topology, it was clear that
the closed interval [a,b] of the real line had a certain property
that was crucial for proving such theorems as the maximum value
theorem and the uniform continuity theorem. But for a long time,
it was not clear how this property should be formulated for an
arbitrary topological space. It used to be thought that the crucial
property of [a,b] was the fact that every infinite subset of
[a,b] has a limit point, and this property was dignified with the
name of compactness. Later, mathematicians realized that this
formulation does not lie at the heart of the matter, but rather a
stronger formulation, in terms of open coverings of the space, is
more central. The latter formulation is what we now call
compactness. It is not as natural or intuitive as the former; some
familiarity with it is needed before is usefulness becomes apparent.
--
Stephen J. Herschkorn sjher...@netscape.net
Math Tutor on the Internet and in Central New Jersey and Manhattan
Paul
A couple more points..
1). Compactness is really emposing a restriction on the number of open sets a space can have.
Also another way to look at it...
2). Compactness is equivalent to : every collection C of closed sets in X having the finite intersection property, the intersection of all elements of C is nonempty.
Finite intersection property: a collection C of subsets of X has F.I.P. if for every finite collection C1,C2,...,Cn of C the intersection C1 /\ C2 /\ ... /\ Cn is nonempty.
James Dolan
linda <zhengxi...@yahoo.com.cn> wrote:
in a non-compact space, there's enough room for "infinite
passing-the-buck" to occur, so that for example a continuous
real-valued function f might never achieve a maximum value. you could
have a dense sequence of points x1,x2,x3,... and each point denies
responsibility for having the maximum value, "passing the buck" on to
the next point in the sequence. x1 tells you "well f(x2) > f(x1) so
go talk to x2" and x2 tells you "f(x3) > f(x2) so go talk to x3" and
so on forever, and the buck never stops; no one ever accepts
responsibility for having the ultimate maximum value.
in a compact space, though, infinite passing-the-buck can't work. all
of that passed-off responsibility starts piling up somewhere and there
has to be a fall guy (a "limit point") to take the ultimate
responsibility if everyone else refuses to accept it. ("the buck
stops here.")
there are alternative ways of formulating the definition of
compactness (or of similar concepts) but they can generally be
interpreted as saying that "no scheme for infinite buck-passing can
work in this space". some versions say something like "every infinite
sequence has a limit point", since an infinite sequence without a
limit point is a sort of scheme for infinite buck-passing. an open
cover without any finite sub-cover can also be thought of as a sort of
scheme for infinite buck-passing; hence the currently most standard
definition of "compact".
--
Gonçalo Rodrigues
fed this fish to the penguins:
Compactness is an elusive notion; you can only master it by looking at
proofs where compactness is used. A good exercise is then to spot
exactly where compactness is used and how the reasoning breaks if we
remove the assumption.
For example, recall that the diameter of a metric space X is defined
as
diam(X) = sup{d(x,y):x,y in X}
where d is the distance.
Proposition: If X is compact it has finite diameter.
proof: Consider the family of open balls B(x,e) centered at x in X and
of radius e >0 (fixed). Since clearly the reunion of these balls
contains X, by compactness there is a *finite* subfamily that covers
X. Since we have a *finite* family of balls of radius, say n balls,
the supremum is bounded by 2ne.
It's obvious that if we do not have compactness, the reasoning fails
miserably. This is a typical technique when it comes to compactness:
you make a construction inside some ball with radius e, then to extend
the construction to the whole space you somehow need compactness to
ensure "finiteness" or "boundedness" or whatever. A case to have in
mind is the proof that continuous functions on compact sets are
uniformly continuous.
Hope it helps, best regards,
G. Rodrigues
jsher
abstraction of the key properties of closed and bounded sets. Picture
the following two sets:
a. the set of points inside a circle, but not including the circle
b. the unit disk - e.g. the circle with everything inside it
What is the key property? (This is what someone refered to as the
Wierstrauss property): If you have a sequence of points in your set
that get arbitrarily close to a certain point (e.g. "converges") then
the limit point is also in the set. In example a, you can get closer
and closer to a point on the boundary of the circle so that your
"limit" is on the circle itself which is not in the set.
Now the concept of "closer" is also abstracted in topology. In metric
spaces, you have a distance measure, so a sequence of points Xn
approaches a limiting point X if:
for every distance , D, then if you go far enough in your sequence you
can find an N'th term so that, every point after N is within distance D
of X.
No matter what distance D your worst enemy picks you can find an N that
works.
In topology, distance measure is replaces by "open sets". A sequence Xn
approaches a limiting point X if for every open set you select, call it
O, that contains X, there is a N'th term so that all points after that
N'th term are also in O.
Basically, metric spaces use open sets of a certain shape (sphere's) to
determine if two points are close together. Other topological spaces
can use a different set of shapes to determine closeness.
Here is the main question: if you have a sequence of points, does it
converge to something in your set? Compactness means that the points
getting closer and closer together is enough to ensure that there is a
limit.
Compact sets have other nice proprties like a continuous function on a
compact set has a maximum and minimum value. So again, they play the
role that closed and bounded sets do in R^n.
Mike Kent
> For example, recall that the diameter of a metric space X is defined
> as
>
> diam(X) = sup{d(x,y):x,y in X}
>
> where d is the distance.
>
> Proposition: If X is compact it has finite diameter.
>
> proof: Consider the family of open balls B(x,e) centered at x in X and
> of radius e >0 (fixed). Since clearly the reunion of these balls
> contains X, by compactness there is a *finite* subfamily that covers
> X. Since we have a *finite* family of balls of radius, say n balls,
> the supremum is bounded by 2ne.
So, the diameter of the two-point subset {0,1} of the real line
is bounded above by 2*2*e for every e>0?
You need a little more: once you have a finite cover {B_i} where
B_i = b(x_i,e), then for any points a and in B_n and B_k, d(ab)
is less than d(x_n, x_k)+2*e. Letting M be max{d{x_i,x_j)} for
1<j<n+1, the diameter is then bounded by M + 2*e.
William Elliot
> 1). Compactness is really emposing a restriction on the number of open
> sets a space can have.
>
The cardinality of the open sets of [0,1] is less
than the cardinality of the open sets of (0,1) ?
Ulysse from CH
<ma...@hevanet.remove.com> wrote:
>> 1). Compactness is really emposing a restriction on the number of open
>> sets a space can have.
>The cardinality of the open sets of [0,1] is less
>than the cardinality of the open sets of (0,1) ?
Just in case this is a real question - and not just an ironic one, as
seems more probable to me: of course the 1st is >= the 2nd, as
there is - by def. of a subspace - always a surjection from the
topology (= set of open sets) of a space to the top. of a subspace.
And as the open interval is homeomorphic to the whole real line,
the converse inequality holds too. so both card. are equal ...
(BTW: because any open set of |R is a disjoint countable union
of open intervals - bounded or not - the card. of |R's top. is that
of |R itself, 2^aleph0.)
Statements similar to "1)" above are often made to communicate that
compactness implies that there are (in a vague sense) 'not too many
open sets' in the space: this is a rather poor method to understand
intuitively the concept, even more if you reduce it to cardinality
(apparently the only possibility to make a precise statement
in this direction) ... Anyhow, 1) isn't supposed to mean or imply
that any compact topology on an arbitrary large set X is "smaller"
than any non-compact topology on any other set Y ! Not even
for Y=X. What 1) *does* mean exactly is though not quite clear
to me (when supposed to be some precise statement); after all,
if the set X underlying a space is fixed, there is an upper bound
for the card. of *any* topology on X: 2^card(X), which is attained
by the discrete top. and ...: if X is infinite (else the discrete X is
already compact) choose a point p in X, make X \ {p} discrete
(so that it is locally compact) and then consider on X the Alexandroff
one-point compactification (p being the point added): this space
has 2^card(X) open sets (any part of X \ {p} is open in X) !
martin dowd
Linda wrote:
>> if you ask me what a compact set is, I can imagine nothing but recite
> its definition awkwardly.
> Can any one help to explain this concept and explain where is is used?
Compactness is a technical concept which generalizes notions found in
specific types of spaces to an arbitrary topological space. It has
become fundamental, although its definition seems mysterious until one
has gained experience with its use.
Some readily understood motivation can be found in Kelley's "General
Topology":
"The Heine Borel theorem asserts that every open cover of a closed and
bounded subset of the space of real numbers has a finite subcover.
Like most good theorems its conclusion has become a definition."
You might try proving the Heine-Borel theorem to gain some insight into
compactness. It is used throughout topology; p. 135 to 156 of Kelley
cover various basic facts.
- Martin Dowd