The fundamental question in reverse mathematics is to determine which set existence axioms are required to prove particular theorems of mathematics. In addition to being interesting in their own right, answers to this question have consequences in both effective mathematics and the foundations of mathematics. Before discussing these consequences, we need to be more specific about the motivating...

This is joint work with Carl Mummert. We initiate the reverse mathematics of general topology. We show that a certain metrization theorem is equivalent to Π 2 comprehension. An MF space is defined to be a topological space of the form MF(P ) with topology generated by {Np | p ∈ P}. Here P is a poset, MF(P ) is the set of maximal filters on P , and Np = {F ∈ MF(P ) | p ∈ F}. If the poset P is co...

A topological space X is Baire if the intersection of any sequence open dense subsets in X. Let $$C_p(X,[0,1])$$ denote all continuous [0, 1]-valued functions on a Tychonoff with topology pointwise convergence. In this paper, we have obtained characterization for function to be separable closed which are $$C^*$$ -embedded. particular, holds normal spaces and, hence, metrizable spaces. Moreover,...

Under the additional assumption of complete regularity, we furnish a simple characterization all topologies such that every continuous total preorder is representable by utility function. In particular, prove completely regular topology satisfies property if, and only it separable linearly ordered collection clopen sets countable. Since not restrictive to refer when dealing with this kind probl...

The space of unitary $C_{0}$-semigroups on separable infinite dimensional Hilbert space, when viewed under the topology uniform weak convergence compact subsets $\mathbb{R}_{+}$, is known to admit various interesting residual subspaces. Before treating contractive case, problem complete metrisability this was raised in [Eisner, 2010]. Utilising Borel complexity computations and automatic contin...

In R d \mathbb {R}^d <...

1. An interesting class of real functions of a single real variable, the approximately continuous functions, was introduced by Denjoy [l] in his work on derivatives. The two principal facts discovered by Denjoy are that these functions are of Baire class 1 and have the Darboux property. Ridder [2] showed that the arguments of Denjoy apply to real functions of n variables. In this paper we discu...

This is joint work with Carl Mummert. We initiate the reverse mathematics of general topology. We show that a certain metrization theorem is equivalent to Π 2 comprehension. An MF space is defined to be a topological space of the form MF(P ) with topology generated by {Np | p ∈ P}. Here P is a poset, MF(P ) is the set of maximal filters on P , and Np = {F ∈ MF(P ) | p ∈ F}. If the poset P is co...

Let PR(X) denote the hyperspace of nonempty finite subsets a topological spaceX with Pixley-Roy topology. In this paper, by introducing closed-miss-finite networks and using principle ultrafilters, we proved that following statements are equivalent for space X: (1) is weakly Rothberger; (2) X satisfies S1(?rcf ,?wrcf ); (3) separable ? {x} S1(?cf ,?wcf ) each x X; (4) principal ultrafilter F [x...

In this work, we are interested in characterizing typical (generic) dimensional properties of invariant measures associated with the full-shift system, $T$, a product space whose alphabet is perfect and separable metric (thus, complete uncountable). More specifically, show that set upper Hausdorff dimension equal to zero lower packing infinity dense $G_\delta$ subset $\mathcal{M}(T)$, $T$-invar...