# Baire space

(redirected from Baire category)

## Baire space

[′ber ‚spās]
(mathematics)
A topological space in which every countable intersection of dense, open subsets is dense in the space.
McGraw-Hill Dictionary of Scientific & Technical Terms, 6E, Copyright © 2003 by The McGraw-Hill Companies, Inc.
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Of course, by means of a Baire category argument (as seen in [8]) one can show that almost every continuous function having zeros has, actually, an uncountable amount of them.
Moreover, we call a set F nowhere dense in E provided C(E\C(F)) = C(E), that is, if C(F) does not contain a nonempty subset of the form E [intersection] w x [X.sup.[omega]], and a subset F is referred to as of first Baire category in E if F is a countable union of sets nowhere dense in E.
Then every regular [omega]-language F [subset or equal to] E is of first Baire category in E if and only if [L.sub.[alpha]](F) = 0.
Moreover, we call a set F nowhere dense in E providedC(E\C(F)) = C(E), that is, if C(F) does not contain a nonempty subset of the form E [intersection] w * [X.sup.[omega]], and a subset F is referred to as of first Baire category in E if F is a countable union of sets nowhere dense in E.
Since E is supposed to be residual in C(E), C(E) \ E is of first Baire category in C(E), and, since E is a countable intersection of regular [omega]-languages, say [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], the difference [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is a countable union of regular [omega]-languages C(E)\[F.sub.i], each of which is of first Baire category in C(E).
Since (q)-porous sets are nowhere dense, all ([sigma], q)-porous sets are of the first Baire category. To point out the difference between (q)-porous and nowhere dense sets, note that if E [subset] Y is nowhere dense, y [member of]Y and r > 0, then there are a point z [member of] Y and a number s > 0 such that B(z,s) [subset] B(y,r) E.
A convex body is typical as soon as it enjoys a property P such that the set of all convex bodies not enjoying P is of the first Baire category in k.
Now, joint continuity of the map follows from Baire category theorem [10].
The latter means that the symmetric difference [U.sub.n] [DELTA][C.sub.n] is meager (i.e., is of the first Baire category) in W.
Although we shared common interest, we have no paper in common, and I followed his work at a distance, mainly through discussions with our common friend Giulio Pianigiani, with whom, at that time, he was developing the Baire Category approach.
is a [G.sub.[delta] dense set by Baire Category Theorem.
Since B is of the second Baire category and the sets ([[Z.sup.n],.sub.m] [intersection] B) are closed, one of these sets would have a nonempty interior.
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