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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...
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.
Let X be a completely regular Hausdorff topological space and the space of continuous real-valued maps on X endowed with the pointwise topology. A simple and natural argument is presented to show how to construct on the space , if X contains a homeomorphic copy of the closed interval [0,1], real-valued maps which are everywhere discontinuous but continuous on all compact subsets of .
In this paper, we prove that a space is a sequentially-quotient -image of a metric space if and only if has a point-star -network consisting of -covers. By this result, we prove that a space is a sequentially-quotient -image of a separable metric space if and only if has a countable -network, if and only if is a sequentially-quotient compact image of a separable metric space; this answers a question raised by Shou Lin affirmatively. We also obtain some results on spaces with countable...
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