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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*.

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Published April 2002.