We consider properties of Tanaka spaces (introduced in Mynard F., More on strongly sequential spaces, Comment. Math. Univ. Carolin. 43 (2002), 525–530), strongly sequential spaces, and weakly sequential spaces. Applications include product theorems for these types of spaces.
Topologically maximal pretopologies, paratopologies and pseudotopologies are characterized in terms of various accessibility properties. Thanks to recent convergence-theoretic descriptions of miscellaneous quotient maps (in terms of topological, pretopological, paratopological and pseudotopological projections), the quotient characterizations of accessibility (in particular, those of G. T. Whyburn and F. Siwiec) are shown to be instances of a single general theorem. Convergence-theoretic characterizations...
A topological space is said to be generated by an ideal if for all and all there is in such that , and is said to be weakly generated by if whenever a subset of contains for every with , then itself is closed. An important class of examples are the so called weakly discretely generated spaces (which include sequential, scattered and compact Hausdorff spaces). Another paradigmatic example is the class of Alexandroff spaces which corresponds to spaces generated by finite sets....
We prove that (A) if a countably compact space is the union of countably many subspaces then it is compact; (B) if a compact space is the union of fewer than = left-separated subspaces then it is scattered. Both (A) and (B) improve results of Tkačenko from 1979; (A) also answers a question that was raised by Arhangel’skiǐ and improves a result of Gruenhage.
A refined common generalization of known theorems (Arhangel’skii, Michael, Popov and Rančin) on the Fréchetness of products is proved. A new characterization, in terms of products, of strongly Fréchet topologies is provided.