43 research outputs found
Two improvements on Tkačenko's addition theorem
summary: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 = \operatorname{cov} (\Cal 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'ski\v{i} and improves a result of Gruenhage
On some classes of Lindel\"of Sigma-spaces
We consider special subclasses of the class of Lindel\"of Sigma-spaces
obtained by imposing restrictions on the weight of the elements of compact
covers that admit countable networks: A space is in the class
if it admits a cover by compact subspaces of weight
and a countable network for the cover. We restrict our attention to
. In the case , the class includes the class
of metrizably fibered spaces considered by Tkachuk, and the -approximable
spaces considered by Tkacenko. The case corresponds to the spaces of
countable network weight, but even the case gives rise to a
nontrivial class of spaces. The relation of known classes of compact spaces to
these classes is considered. It is shown that not every Corson compact of
weight is in the class , answering a question
of Tkachuk. As well, we study whether certain compact spaces in
have dense metrizable subspaces, partially answering a
question of Tkacenko. Other interesting results and examples are obtained, and
we conclude the paper with a number of open questions.Comment: 21 pages. to appear in Topology and its Application
Some new directions in infinite-combinatorial topology
We give a light introduction to selection principles in topology, a young
subfield of infinite-combinatorial topology. Emphasis is put on the modern
approach to the problems it deals with. Recent results are described, and open
problems are stated. Some results which do not appear elsewhere are also
included, with proofs.Comment: Small update
On a condition for the pseudo radiality of a product
summary:A sufficient condition for the pseudo radiality of the product of two compact Hausdorff spaces is given
On compact fibered spaces
AbstractA space X is called fibered if there exists a countable family γ of sets closed in X such that γ(x)=⋂{F:x∈F∈γ} is metrizable for each x∈X. In the paper we answer two problems of Tkachuk raised in [Topology Proc. 19 (1994) 321–334] about compact fibered spaces
Some properties of C(X), I
AbstractBy a result of A.V. Arhangel'skiǐ and E.G. Pytkeiev, the space C(X) of the continuous real functions on X with the topology of pointwise convergence has tightness ω iff Xn is Lindelöf for every n ∈ ω. In this paper we describe other convergence properties of C(X) (e.g. the Fréchet-Urysohn properly) in terms of covering properties of X.In some cases the equivalence between these properties turn out to be dependent on the set theory we choose. Some open problems are also stated