Topology Atlas Document # ppae-10


Characterizing continuity by preserving compactness and connectedness

János Gerlits, István Juhász, Lajos Soukup and Zoltán Szentmiklóssy

Proceedings of the Ninth Prague Topological Symposium (2001) pp. 93-118

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 Frèchet, 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\kappa) and f:X --> Y is a preserving function into a regular space Y, then f is continuous.

Mathematics Subject Classification. 54C05 54D05 54F05 54B10.
Keywords. Hausdorff space, continuity, compact, connected, locally connected, Frechetspace, monotonically normal, linearly ordered space.

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Comments. This article has been submitted for publication to Fundamenta Mathematicae.

Copyright © 2002 Charles University and Topology Atlas. Published April 2002.