Characterizing continuity by preserving compactness and connectedness

2002/04/10 by Janos Gerlits, Istvan Juhasz, Lajos Soukup +1
Mathematics · #math.GN #msc:54C05 #msc:54D05 #msc:54F05 #msc:54B10

paper · pdf

published as Proceedings of the Ninth Prague Topological Symposium, (Prague, 2001), pp. 93--118, Topology Atlas, Toronto, 2002 · 26 pages. This article has been submitted for publication to Fundamenta Mathematicae

arxiv created 2002/04/10 · arxiv updated 2009/11/30

Abstract

Let us call a function f from a space X into a space Y preserving if the image of every compact subspace of X is compact in Y and the image of every connected subspace of X is connected in Y. By elementary theorems a continuous function is always preserving. Evelyn R. McMillan proved in 1970 that if X is Hausdorff, locally connected and Frechet, Y is Hausdorff, then the converse is also true: any preserving function f:X→ Y is continuous. The main result of this paper is that if X is any product of connected linearly ordered spaces (e.g. if X = R^κ) and f:X → Y is a preserving function into a regular space Y, then f is continuous.

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