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- W4290851463 abstract "A function $f:Xto Y$ between topological spaces is called {em compact-preserving} if the image $f(K)$ of each compact subset $Ksubset X$ is compact. We prove that a function $f:Xto Y$ defined on a strong Frechet space $X$ is compact-preserving if and only if for each point $xin X$ there is a compact subset $K_xsubset Y$ such that for each neighborhood $O_{f(x)}subset Y$ of $f(x)$ there is a neighborhood $O_xsubset X$ of $x$ such that $f(O_x)subset O_{f(x)}cup K_x$ and the set $K_xsetminus O_{f(x)}$ is finite. This characterization is applied to give an alternative proof of a classical characterization of continuous functions on locally connected metrizable spaces as functions that preserve compact and connected sets. Also we show that for each compact-preserving function $f:Xto Y$ defined on a (strong) Fr'echet space $X$, the restriction $f|LI'_f$ (resp. $f|LI_f)$ is continuous. Here $LI_f$ is the set of points $xin X$ of local infinity of $f$ and $LI'_f$ is the set of non-isolated points of the set $LI_f$. Suitable examples show that the obtained results cannot be improved." @default.
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- W4290851463 date "2012-08-11" @default.
- W4290851463 modified "2023-10-18" @default.
- W4290851463 title "The continuity properties of compact-preserving functions" @default.
- W4290851463 doi "https://doi.org/10.48550/arxiv.1208.2319" @default.
- W4290851463 hasPublicationYear "2012" @default.
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