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Sequential compactness vs. countable compactness

Angelo Bella, Peter Nyikos (2010)

Colloquium Mathematicae

The general question of when a countably compact topological space is sequentially compact, or has a nontrivial convergent sequence, is studied from the viewpoint of basic cardinal invariants and small uncountable cardinals. It is shown that the small uncountable cardinal 𝔥 is both the least cardinality and the least net weight of a countably compact space that is not sequentially compact, and that it is also the least hereditary Lindelöf degree in most published models. Similar results, some definitive,...

Sequential + separable vs sequentially separable and another variation on selective separability

Angelo Bella, Maddalena Bonanzinga, Mikhail Matveev (2013)

Open Mathematics

A space X is sequentially separable if there is a countable D ⊂ X such that every point of X is the limit of a sequence of points from D. Neither “sequential + separable” nor “sequentially separable” implies the other. Some examples of this are presented and some conditions under which one of the two implies the other are discussed. A selective version of sequential separability is also considered.

Short proofs of two theorems in topology

Mohammad Ismail, Andrzej Szymański (1993)

Commentationes Mathematicae Universitatis Carolinae

We present short and elementary proofs of the following two known theorems in General Topology: (i) [H. Wicke and J. Worrell] A T 1 weakly δ θ -refinable countably compact space is compact. (ii) [A. Ostaszewski] A compact Hausdorff space which is a countable union of metrizable spaces is sequential.

Spaces in which compact subsets are closed and the lattice of T 1 -topologies on a set

Ofelia Teresa Alas, Richard Gordon Wilson (2002)

Commentationes Mathematicae Universitatis Carolinae

We obtain some new properties of the class of KC-spaces, that is, those topological spaces in which compact sets are closed. The results are used to generalize theorems of Anderson [1] and Steiner and Steiner [12] concerning complementation in the lattice of T 1 -topologies on a set X .

Strongly sequential spaces

Frédéric Mynard (2000)

Commentationes Mathematicae Universitatis Carolinae

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.

Tanaka spaces and products of sequential spaces

Yoshio Tanaka (2007)

Commentationes Mathematicae Universitatis Carolinae

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 convergences, accessibility, and covering maps

Szymon Dolecki, Michel Pillot (1998)

Mathematica Bohemica

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...

Currently displaying 101 – 120 of 133