What is meant by a meager subset??
Steven
Robert Israel
is contained in the union of countably many closed
nowhere-dense sets.
Robert Israel isr...@math.ubc.ca
Department of Mathematics http://www.math.ubc.ca/~israel
University of British Columbia Vancouver, BC, Canada
Chip Eastham
interior.
A meager subset (or meagre, for those across the pond) is a countable
union of nowhere dense subsets.
These topological concepts tie in with that of Baire category.
regards, chip
George Cox
>
> Can some please define a meager subset for me?
A meagre set (in a topological space) is the same as a set of first
category. I.e. a set that is the union of at most aleph_0 nowhere dense
sets.
A nowhere dense set is one with the interior of its closure equal to the
empty set.
G. A. Edgar
terms, so he invented some new ones: meagre, residual. Do you think
they are better terms?
--
G. A. Edgar http://www.math.ohio-state.edu/~edgar/
Dave L. Renfro
[sci.math Dec 20 2004 9:26:17:000AM]
http://mathforum.org/discuss/sci.math/m/664050/664247
wrote:
> Bourbaki did not think "first category" and
> "second category" were good terms, so he
> invented some new ones: meagre, residual.
> Do you think they are better terms?
Actually, Denjoy introduced the term "residual"
around 1912 or 1913. I believe he was trying to
distinguish between a set being second category
(i.e. the set is big) and the set having a first
category complement (i.e. the set is so big that
what's left over is small). During this period some
authors (most people, including Baire and Lebesgue)
used "second category" for "not first category",
while other authors (Lusin, Schönflies, Hobson)
used "second category" for "residual". Then, to
make matters even worse, when "second category"
meant "not first category" for an author, quite
often that author would use second category in the
statement of a theorem, but then wind up proving
the set was actually residual. (This last thing
still happens, especially in areas of math that
are not very close to topology or analysis.)
The term "generic" is also used for "residual",
but I've come across some instances (I think the
papers were in dynamical systems theory and/or
differential geometry) where an author uses
"generic" to mean "open dense set" (i.e. the complement
is not just first category, it's actually nowhere
dense), or perhaps more generally, "contains an open
dense set".
Dave L. Renfro
Mike Oliver
> G. A. Edgar <ed...@math.ohio-state.edu.invalid>
> [sci.math Dec 20 2004 9:26:17:000AM]
> http://mathforum.org/discuss/sci.math/m/664050/664247
> wrote:
>>Bourbaki did not think "first category" and
>>"second category" were good terms, so he
>>invented some new ones: meagre, residual.
>>Do you think they are better terms?
>
>
> Actually, Denjoy introduced the term "residual"
> around 1912 or 1913. I believe he was trying to
> distinguish between a set being second category
> (i.e. the set is big) and the set having a first
> category complement (i.e. the set is so big that
> what's left over is small).
So the contemporary term for the second concept
is "comeager". It has the advantage of providing
a parallel term "conull" for a set whose complement
is null. "Null" means "measure zero" in whatever
measure is currently being considered, or more generally,
that the set is an element of whatever ideal is currently
under consideration.
> The term "generic" is also used for "residual",
This use of "generic" matches up nicely with forcing
terminology, but rather than saying a big set is "generic",
you more usually say that something happens "generically often".
Dave L. Renfro
Date Posted: Dec 20 2004 6:26:21:000PM
http://mathforum.org/discuss/sci.math/m/664050/664408
wrote (in part):
> So the contemporary term for the second concept
> is "comeager". It has the advantage of providing
> a parallel term "conull" for a set whose complement
> is null. "Null" means "measure zero" in whatever
> measure is currently being considered, or more generally,
> that the set is an element of whatever ideal is currently
> under consideration.
In the past few years I've grown fond of using the
prefix "co" as in co-meagerly, co-measure zero,
co-countable, etc. A lot of the kinds of results
I'm interested in can be described nicely using
quantifiers modulo some notion of smallness:
There exist co-meagerly many bounded derivatives
from [0,1] into R (sup norm) that are discontinuous
at co-measure zero many points, (Weil's Zbl 377.26005),
and every bounded derivative from [0,1] into R is
continuous at co-meagerly many points (Baire, 1899).
For co-meagerly many compact subsets E of [0,1]
(Hausdorff metric) and for co-meagerly many
continuous measures mu on [0,1] (weak * topology
via Riesz, I think), mu(E) = 0. (Dubins/Freedman's
MR 30 # 4887, 3.11 and 3.12 on p. 1216)
If C is a closed nowhere dense subset of R, then
at co-meagerly many points of C (meager relative to C)
both the left and right lower Lebesgue densities are
zero and at co-measure zero many points of C (usual
Lebesgue measure in R) both the left and the right
lower Lebesgue densities are one. (Denjoy's MR 8,260i,
pp. 195-196 for first half; Lebesgue density theorem
for second half)
For co-meagerly many continuous f:[0,1] --> R
(sup norm), we have d- = d+ = -oo and D- = D+ = oo
(lower and upper Dini derivates of f) at co-meagerly
many and co-measure zero many points, and we have
d- = D- = oo at c-densely (in R) many points.
(Jarnik's Zbl 7.40102 and Saks's Zbl 5.39105)
Incidentally, Alexander Kechris makes use of cardinality,
category, and measure versions of "there exists" and
"for all" in his book "Classical Descriptive Set Theory".
I think what he does there leads to a neat way of
stating these kinds of results.
Dave L. Renfro