Reflecting topological properties in continuous images
Given a topological property P, we study when it reflects in small continuous images, i.e., when for some infinite cardinal κ, a space X has P if and only if all its continuous images of weight less or equal to κ have P. We say that a cardinal invariant η reflects in continuous images of weight κ + if η(X) ≤ κ provided that η(Y) ≤ κ whenever Y is a continuous image of X of weight less or equal to κ +. We establish that, for any infinite cardinal κ, the spread, character, pseudocharacter and Souslin...
Reflexive families of closed sets
Let S(X) denote the set of all closed subsets of a topological space X, and C(X) the set of all continuous mappings f:X → X. A family 𝓐 ⊆ S(X) is called reflexive if there exists ℱ ⊆ C(X) such that 𝓐 = {A ∈ S(X): f(A) ⊆ A for every f ∈ ℱ}. We investigate conditions ensuring that a family of closed subsets is reflexive.
Relations approximated by continuous functions in the Vietoris topology
Let X be a Tikhonov space, C(X) be the space of all continuous real-valued functions defined on X, and CL(X×ℝ) be the hyperspace of all nonempty closed subsets of X×ℝ. We prove the following result: Let X be a locally connected locally compact paracompact space, and let F ∈ CL(X×ℝ). Then F is in the closure of C(X) in CL(X×ℝ) with the Vietoris topology if and only if: (1) for every x ∈ X, F(x) is nonempty; (2) for every x ∈ X, F(x) is connected; (3) for every isolated x ∈ X, F(x) is a singleton...
Remarks on extremally disconnected semitopological groups
Answering recent question of A.V. Arhangel'skii we construct in ZFC an extremally disconnected semitopological group with continuous inverse having no open Abelian subgroups.
Remarks on Namioka spaces and R. E. Johnson's theorem on the norm seperability of the range of certain mappings.
Remarks on topologies uniquely determined by their continuous self maps
Representability of concrete categories by non-constant morphisms