We give a straightforward topological description of a class of spaces that are separable, countably compact, countably tight and Urysohn, but not compact or sequential. We then show that this is the same class of spaces constructed by Manes [Monads in topology, Topology Appl. 157 (2010), 961--989] using a category-theoretical framework.
For a non-isolated point of a topological space let be the smallest cardinality of a family of infinite subsets of such that each neighborhood of contains a set . We prove that (a) each infinite compact Hausdorff space contains a non-isolated point with ; (b) for each point with there is an injective sequence in that -converges to for some meager filter on ; (c) if a functionally Hausdorff space contains an -convergent injective sequence for some meager filter...
The aim of this paper is to study some properties of Michálek’s fuzzy topology which are quite different of the classic properties of the Chang’s topology.
In this article, we extend the work on minimal Hausdorff functions initiated by Cammaroto, Fedorchuk and Porter in a 1998 paper. Also, minimal Urysohn functions are introduced and developed. The properties of heredity and productivity are examined and developed for both minimal Hausdorff and minimal Urysohn functions.
In this article we introduce the notion of strongly -spaces, that is, those spaces in which countably compact subsets are closed. We find they have good properties. We prove that a space is maximal countably compact if and only if it is minimal strongly , and apply this result to study some properties of minimal strongly -spaces, some of which are not possessed by minimal -spaces. We also give a positive answer to a question proposed by O. T. Alas and R. G. Wilson, who asked whether every...
An -space is a topological space in which the topology is generated by the family of all -sets (see [N]). In this paper, minimal--spaces (where denotes several separation axioms) are investigated. Some new characterizations of -spaces are also obtained.
Some results concerning spaces with countably weakly uniform bases are generalized for spaces with -in-countable ones.
A subset of a product of topological spaces is called -thin if every its two distinct points differ in at least coordinates. We generalize a construction of Gruenhage, Natkaniec, and Piotrowski, and obtain, under CH, a countable space without isolated points such that contains an -thin dense subset, but does not contain any -thin dense subset. We also observe that part of the construction can be carried out under MA.
The minimum weight of a nowhere first-countable compact space of countable -weight is shown to be , the least cardinal for which the real line can be covered by many nowhere dense sets.