Characterizing continuity by preserving compactness and connectedness

Details Characterizing continuity by preserving compactness and connectedness Characterizing continuity by preserving compactness and connectedness

2002-04-10

arxiv.org/abs/math/0204125v1

Proceedings of the Ninth Prague Topological Symposium, (Prague, 2001), pp. 93--118, Topology Atlas, Toronto, 2002

Mathematics

General Topology

26 pages. This article has been submitted for publication to Fundamenta Mathematicae

Scientific paper

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 $f:X\to Y$ is continuous. The main result of this paper is that if $X$ is any product of connected linearly ordered spaces (e.g. if $X = R^\kappa$) and $f:X \to Y$ is a preserving function into a regular space $Y$, then $f$ is continuous.

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