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Topologies generated by ideals

Carlos Uzcátegui (2006)

Commentationes Mathematicae Universitatis Carolinae

A topological space X is said to be generated by an ideal ℐ if for all A ⊆ X and all x ∈ A ¯ there is E ⊆ A in ℐ such that x ∈ E ¯ , and is said to be weakly generated by ℐ if whenever a subset A of X contains E ¯ for every E ⊆ A with E ∈ ℐ , then A 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....

Two improvements on Tkačenko's addition theorem

János Gerlits, István Juhász, Zoltán Szentmiklóssy (2005)

Commentationes Mathematicae Universitatis Carolinae

We prove that (A) if a countably compact space is the union of countably many D subspaces then it is compact; (B) if a compact T 2 space is the union of fewer than N ( ℝ ) = cov ( ℳ ) 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.

Two-fold theorem on Fréchetness of products

Szymon Dolecki, Tsugunori Nogura (1999)

Czechoslovak Mathematical Journal

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.

Weak-open compact images of metric spaces

Sheng Xiang Xia (2008)

Czechoslovak Mathematical Journal

The main results of this paper are that (1) a space X is g -developable if and only if it is a weak-open π image of a metric space, one consequence of the result being the correction of an error in the paper of Z. Li and S. Lin; (2) characterizations of weak-open compact images of metric spaces, which is another answer to a question in in the paper of Y. Ikeda, C. liu and Y. Tanaka.

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