A. Szymanski

Closed graph theorem: Topological approach

Some new types of Closed Graph Theorem are presented. These results generalize some theorems of T. Byczkowski, R. Pol and M. Wilhelm. An answer to a problem of M. Wilhelm is provided.

On the metrizability number and related invariants of spaces, II

Abstract This is a continuation of the study of the metrizability number and the first countability number for various classes of compact Hausdorff spaces started by Ismail and Szymanski (1995). It is shown that if X is a compact LOTS, then w ( X ) ⩽ ω · m ( X ). Also, if X is the one-point compactification of an uncountable discrete space, then ω 1 ⩽ m ( X ω ) ⩽ 2 ω . Furthermore, under the singular cardinals hypothesis, for a large class of spaces of cardinality > 2 ω , the first countability number, the metrizability number and the cardinality coincide.

Compact spaces representable as unions of nice subspaces

We present several results on compact Hausdorff spaces which can be represented as unions of nice subspaces. Some typical results are: If X is a compact Hausdorff space, and X = ∪α κ and μ is regular, then there exists a discrete sequence {xα: α<μ} in X such that xα→x, (iii) if A is a nonclosed subset of X, then there exists a point x ϵ X\A and a filter base F of subsets of A such that | F | ⩽κ and F→x. We also show that if a compact Hausdorff space X is a union of countably many metrizable spaces, X has no isolated points and c(X)=ω0, then X is a compactification of the space of irrationals.

Some remarks on Park's abstract convex spaces

We discuss S. Parks abstract convex spaces and their relevance to classical convexieties and

On locally compact Hausdorff spaces with finite metrizability number

L^{\ast }

On subspaces of exp(N)

-operators. We construct an example of a space satisfying the partial KKM principle that is not a KKM space. The existence of such a space solves a problem by S. Park.

Q-binary spaces

Abstract The metrizability number m(X) of a space X is the smallest cardinal number κ such that X can be represented as a union of κ many metrizable subspaces. In this paper, we study compact Hausdorff spaces with finite metrizability number. Our main result is the following representation theorem: If X is a locally compact Hausdorff space with m(X)=n , then for each k , 1⩽k , X can be represented as X=G∪F , where G is an open dense subspace, F=X⧹G , m(G)=k , and m(F)=n−k .

Spaces hereditarily of κ-type and point κ-type

Let exp(X) denote the exponential space of a topological spaceX introduced by Vietoris [14]. In this paper, we study the subspaces of the space exp(N), whereN={1, 2, 3, ...} is the discrete space of natural numbers. We also show that if the metrizability number of exp(X) is countable, thenX (and exp(X)) must be compact and metrizable.

Proper Functions with the Baire Property

Abstract We distinguish and investigate the class of Q -binary spaces. Several equivalent descriptions enable us to show a rich geometric structure of Q -binary connected spaces. They coincide, up to homeomorphism, with spaces that admit binary normal closed subbase.

Applications of General Infimum Principles to Fixed-Point Theory and Game Theory

In this paper we discuss the structure of regular spaces hereditarily of κ-type and point κ-type. In particular, it is shown that a regular spaceX is hereditarily of point κ-type iff for everyp∈X there is a setE (possibly empty) of isolated points ofX such that (1)E∪{p} is compact and (2)X(E∪{p}, X)≤κ.