Frechet-Urysohn Property Research Articles

On properties of infimal topology of a map space

We study properties of the infimal topology τinf which is the infimum of the family of all topologies of uniform convergence defined on the set C(X, Y) of continuous maps into a metrizable space Y. One of the main results of the research consists in obtaining necessary and sufficient condition for the topology τinf to have the Frechet–Urysohn property. We also establish necessary and sufficient conditions for coincidence of the topology τinf and a topology of uniform convergence τμ (“attaining” the infimum). We prove that for this coincidence it is sufficient for the topology τinf to satisfy the first axiom of countability.

Discrete reflexivity and complements of the diagonal

We prove that, for a countably compact space X of weight at most ω1, if \({\mathcal {P}}\) ∈{countable tightness, Frechet–Urysohn property, sequentiality, first countability} and the closure of every discrete subspace of X has \({\mathcal {P}}\) then X has \({\mathcal {P}}\). Given a countably compact space X, if X2∖Δ is discretely Lindelof then X is metrizable. We establish that a Lindelof Σ-space X is cosmic whenever X2∖Δ is paracompact; this generalizes the respective result of Gruenhage for compact X. Furthermore, a countably compact space X is metrizable if the closure of every discrete subspace of X2 is metrizable. It turns out that many non-discretely reflexive properties behave much better in finite powers of spaces. In particular, if X is compact and \(\overline{D}\) is Corson (Eberlein) compact for any discrete D⊂X3 then X is a Corson (Eberlein) compact.

Metrizability and the Frechet-Urysohn Property in Topological Groups

Metrizability and the Frechet-Urysohn Property in Topological Groups