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 $G(X)$....

A notion of resource-bounded Baire category is developed for the class PC[0,1] of all polynomial-time computable real-valued functions on the unit interval. The meager subsets of PC[0,1] are characterized in terms of resource-bounded Banach-Mazur games. This characterization is used to prove that, in the sense of Baire category, almost every function in PC[0,1] is nowhere differentiable. This i...

A function from Baire space N to the natural numbers N is called formally continuous if it is induced by a morphism between the corresponding formal spaces. We compare formal continuity to two other notions of continuity on Baire space working in Bishop constructive mathematics: one is a function induced by a Brouwer-operation (i.e. inductively defined neighbourhood function); the other is a fu...

Using ♦ and large cardinals we extend results of Magidor–Malitz and Farah–Larson to obtain models correct for the existence of uncountable homogeneous sets for finite-dimensional partitions and universally Baire sets. Furthermore, we show that the constructions in this paper and its predecessor can be modified to produce a family of 2ω1 -many such models so that no two have a stationary, costat...

We show that a real sequence x is convergent if and only there exist regular matrix A an Fσδ-ideal I on N such the set of subsequences y for which Ay I-convergent second Baire category. This includes cases where ideal asymptotic density zero sets, Banach finite sets. The latter recovers old result given by Keogh Petersen in (1958) [26]. Our proofs are different nature rely recent results contex...

in this paper, we introduce the cone normed spaces and cone bounded linear mappings. among other things, we prove the baire category theorem and the banach--steinhaus theorem in cone normed spaces.