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Relations approximated by continuous functions in the Vietoris topology

L'. Holá, R. A. McCoy (2007)

Fundamenta Mathematicae

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

Igor V. Protasov (2002)

Commentationes Mathematicae Universitatis Carolinae

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.

Some remarks providing discontinuous maps on some C p ( X ) spaces

S. Moll (2008)

Banach Center Publications

Let X be a completely regular Hausdorff topological space and C p ( X ) 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 C p ( X ) , 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 C p ( X ) .

Spaces with countable s n -networks

Ge Ying (2004)

Commentationes Mathematicae Universitatis Carolinae

In this paper, we prove that a space X is a sequentially-quotient π -image of a metric space if and only if X has a point-star s n -network consisting of c s * -covers. By this result, we prove that a space X is a sequentially-quotient π -image of a separable metric space if and only if X has a countable s n -network, if and only if X 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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