A space X is a D-space if whenever one is given a neighborhood N(x) of x for each x ∈ X, then there is a closed discrete subset D of X such that {N(x) : x ∈ D} covers X. It is a decades-old open question whether some of the standard covering properties, such as Lindelöf or paracompact, imply D . This remains unsettled, yet there has been a considerable amount of interesting recent work on D-spa...

The axiom of multiple choice implies that metric spaces are paracompact but the reverse implication cannot be proved in set theory without the axiom of choice. 1. Background, Definitions and Summary of Results. Working in set theory without the axiom of choice we study the deductive strength of the assertion MP: Metric spaces are paracompact. (Definitions are given below.) MP was first proved i...

An S-space is a normal topological space in which each covering by open sets has a refinement which is star-finite, that is, each set of the refinement meets only a finite number of sets of the refinement. Thus a compact ( = bicompact) space is an S-space, and an S-space is paracompact [ l ] . 1 In this note we discuss cartesian products in which one of the factors is an S-space. We show that i...

In this paper we continue with the study of paracompact maps introduced in [1]. We give two external characterizations for paracompact maps including a characterization analogous to The Tamano Theorem in the category TOP (of topological spaces and continuous maps as morphisms). A necessary and sufficient condition for the Tychonoff product of a closed map and a compact map to be closed is also ...

In this paper we define a space σ(X) for approximate systems of compact spaces. The construction is due to H. Freudenthal for usual inverse sequences [4, p. 153–156]. We stablish the following properties of this space: (1) The space σ(X) is a paracompact space, (2) Moreover, if X is an approximate sequence of compact (metric) spaces, then σ(X) is a compact (metric) space (Lemma 2.4). We give th...

For a topological space X, let L(X) be the modal logic of X where is interpreted as interior (and hence ♦ as closure) in X. It was shown in [6] that the modal logics S4, S4.1, S4.2, S4.1.2, S4.Grz, S4.Grzn (n ≥ 1), and their intersections arise as L(X) for some Stone space X. We give an example of a scattered Stone space whose logic is not such an intersection. This gives an affirmative answer ...

Definition of bi-supra topological space via graph theory was introduced in this study. We also studied some concepts related bi supra-topological theory. "At least many theorems were proofed as a characterization and examples to explain the subject".