Baire space


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Baire space

[′ber ‚spās]
(mathematics)
A topological space in which every countable intersection of dense, open subsets is dense in the space.
References in periodicals archive ?
A Banach space X is a Stegall space if for every Baire space Z, every usco minimal function f: Z [right arrow] X' (with the weak topology) is single-valued in a residual subset.
We say (U, E, [psi]) is a fuzzy soft Baire space if each sequence [mathematical expression not reproducible] of fuzzy soft nowhere dense sets in (U, E, [psi]) such that [mathematical expression not reproducible].
Then (X, T) is said to neutrosophic Baire space if Nint([[union].sup.[infinity].sub.i=1] [A.sub.i]) = [0.sub.N], where [A.sub.i]'s are neutrosophic nowhere dense sets in (X, T).
But [OMEGA]* being weakly compact, it is a Baire space. Therefore there is [??] [member of] [??] such that B[ [??], [epsilon]/2] [intersection] [OMEGA]* has non-void interior in the relative weak topology.
Gregoriades examines the equivalence classes under effective Borel isomorphism, between complete separable metric spaces that admit a recursive presentation, and show the existence of strictly increasing and strictly decreasing sequences as well as of infinite anti-chains under the natural notion of effective isomorphism reduction, as opposed to the non-effective case, where only two such classes exist, the one of the Baire space and the one of the naturals.
If X is a Baire space and Y is metrizable, then the set of universal vectors for [{[L.sub.n]}.sub.n] is residual in X if and only if the set {(x, [L.sub.n] x) : x [member of] X, n [member of] N} is dense in X x Y.
If X is a vg-compact vg-[R.sub.1]-space, then X is a Baire space.
Let K be the Baire space of all convex bodies in [R.sup.3].
Then (X, T) is called a Fuzzy Baire space if int ([[disjunction].sup.[infinity].sub.i = 1]([[lambda].sub.i])) = 0, where [[lambda].sub.i]'s are Fuzzy nowhere dense sets in (X, T).
First of all, let us recall that the group Sym(X) of permutations of X endowed with the topology of pointwise convergence ([double dagger]) is a Baire space. Recall that a subset Y [subset] Sym(X) is meagre if it is a union of countably many closed subsets with empty interior; and generic or dense [G.sub.[delta]] if its complement Sym(X) \ Y is meagre.
Moreover observe that D is dense in X, in fact every tr([f.sub.n]) is a dense [G.sub.[delta]-set and therefore D is the intersection of a countable family of open dense sets in the Baire space X.
Before Kelly, bitopological space appeared in a narrow sense in [25] as a supplementary work to characterize Baire spaces. In 1990, Ivanov [26] presented a new viewpoint for the theory of bitopological spaces by using a topologic structure on the cartesian product of two sets.