It will be shown that if in a family of sets there exists a strong sequence of the length then this family contains a subfamily consisting of pairwise disjoint sets. The method of strong sequences will be used for estimating the weight of regular spaces.
One of the most important and well known theorem in the class of dyadic spaces is Esenin-Volpin's theorem of weight of dyadic spaces. The aim of this paper is to prove Esenin-Volpin's theorem in general form in class of thick spaces which possesses special subbases.
J. Keesling has shown that for connected spaces the natural inclusion of in its Stone-Čech compactification is a shape equivalence if and only if is pseudocompact. This paper establishes the analogous result for strong shape. Moreover, pseudocompact spaces are characterized as spaces which admit compact resolutions, which improves a result of I. Lončar.
A space is said to be strongly base-paracompact if there is a basis for with such that every open cover of has a star-finite open refinement by members of . Strongly paracompact spaces which are strongly base-paracompact are studied. Strongly base-paracompact spaces are shown have a family of functions with cardinality equal to the weight such that every open cover has a locally finite partition of unity subordinated to it from .
The concept of a strongly chaotic space is introduced, and its relations to chaotic, rigid and strongly rigid spaces are studied. Some sufficient as well as necessary conditions are shown for a dendrite to be strongly chaotic.
Strongly paracompact metrizable spaces are characterized in terms of special S-maps onto metrizable non-Archimedean spaces. A similar characterization of strongly metrizable spaces is obtained as well. The approach is based on a sieve-construction of "metric"-continuous pseudo-sections of lower semicontinuous mappings.
The problem of Y. Tanaka [10] of characterizing the topologies whose products with each first-countable space are sequential, is solved. The spaces that answer the problem are called strongly sequential spaces in analogy to strongly Fréchet spaces.