Characterizing continuity by preserving compactness and connectedness

János Gerlits, István Juhász, Lajos Soukup and Zoltán Szentmiklóssy

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\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={\mathbb{R}}^\kappa$) and $ f:X\to Y$ is a preserving function into a regular space $ Y$, then $ f$ is continuous.

Key words and phrases: Hausdorff space, continuity, compact, connected, locally connected, Fr\`echet space, monotonically normal, linearly ordered space

2000 Mathematics Subject Classification: 54C05, 54D05, 54F05, 54B10

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