The aim of this paper is to study some properties of soft Hausdorff space introduced by Shabir and Naz. Firstly, we give a representation of soft sets and soft topological spaces. Secondly, we introduce some new concepts in soft topological space such as convergence of sequences, homeomorphism and investigate the relations between these concepts and Hausdorff axiom in soft topological space. 20...

We develop the multifractal analysis of conformal axiom A flows. This includes the study of the Hausdorff dimension of basic sets of the flow, the description of the dimension spectra for pointwise dimension and for Lyapunov exponents and the multifractal decomposition associated with these spectra. The main tool of study is the thermodynamic formalism for hyperbolic flows by Bowen and Ruelle. ...

A general combinatorial theorem for countably compact, noncompact spaces is given under the Proper Forcing Axiom. It follows that compact Hausdorff spaces of countable tightness are sequential under PFA, solving the Moore-Mrowka Problem. Other applications are also given.

In ZF (i.e., Zermelo-Fraenkel set theory without the Axiom of Choice) the following statements are shown to be equivalent: (1) The axiom of dependent choice. (2) Products of compact Hausdorff spaces are Baire. (3) Products of pseudocompact spaces are Baire. (4) Products of countably compact, regular spaces are Baire. (5) Products of regular-closed spaces are Baire. (6) Products of Čech-complete...

Assuming Jensen’s principle ♦, there is a compact Hausdorff space X which is hereditarily Lindelöf, hereditarily separable, and connected, such that no closed subspace of X is both perfect and totally disconnected. The Proper Forcing Axiom implies that there is no such space. The ♦ example also fails to satisfy the CSWP (the complex version of the Stone-Weierstrass Theorem). This space cannot c...

In the first part of this thesis (Chapter 1), we will identify ultrafilters on ω with subspaces of 2 through characteristic functions, and study their topological properties. More precisely, let P be one of the following topological properties. • P = being completely Baire. • P = countable dense homogeneity. • P = every closed subset has the perfect set property. We will show that, under Martin...

This is a continuation of [dhhkr]. We study the Tychonoff Compactness Theorem for various definitions of compact and for various types of spaces, (first and second countable spaces, Hausdorff spaces, and subspaces of R). We also study well ordered Tychonoff products and the effect that the multiple choice axiom has on such products.

The bitopological unstability of RR-pairwise paracompactness in presence of pairwise Hausdorff separation axiom is caused by a bitopological property which is much weaker and more local than RR-pairwise paracompactness. We slightly generalize some Michael’s constructions and characterizeRR-pairwise paracompactness in terms of bitopological θ-regularity, and some other weaker modifications of pa...