The article studies topological games that arise in the study of continuity operations groups with topology, such as paratopological and semitopological groups. These are modifications Banach--Mazur game. Given a two-player game $G(X)$ type, we define $\Gamma^G$-Baire, $\Gamma^G$-nonmeager $\Gamma^G$-spaces. A space $X$ is $\Gamma^G$-Baire if second player does not have winning strategy $G(X)$....

The investigation of computational properties of discontinuous functions is an important concern in computable analysis. One method to deal with this subject is to consider effective variants of Borel measurable functions. We introduce such a notion of Borel computability for single-valued as well as for multi-valued functions by a direct effectivization of the classical definition. On Baire sp...

We prove the following theorem: THEOREM. Let Y be a second countable, infinite R0-space. If there are countably many open sets 01, 02, 0n, in Y such that 01 02 0..., then a topological space X is a Baire space if and only if every mapping f: XY is almost continuous on a dense subset of X. It is an improvement of a theorem due to Lin and Lin [2].

The transitive closure of a reflexive, symmetric, analytic relation is an analytic equivalence relation. Does some smaller class contain the transitive closure of every reflexive, symmetric, closed relation? An essentially negative answer is provided here. Every analytic equivalence relation on an arbitrary Polish space is Borel embeddable in the transitive closure of the union of two smooth Bo...

Assume that X is a Banach space such that its Szlenk index Sz X is less than or equal to the first infinite ordinal ω. We prove that X can be renormed in such a way that X with the resultant norm satisfies R X < 2, where R · is the Garcı́a-Falset coefficient. This leads us to prove that if X is a Banach space which can be continuously embedded in a Banach space Y with Sz Y ≤ ω, then, X can be re...

The Baire category theorem implies that the family, F,of dense sets G6 in a fixed metric space, X , is a candidate for generic sets since it is closed under countable intersections; and if X is perfect (has no isolated point), then A E F has uncountable intersections with any open ball in X. There is a long tradition of soft arguments to prove that certain surprising sets are generic. For examp...