Inverse cluster sets

Hamlett, T. R.; Long, Paul

doi:10.1090/S0002-9939-1975-0388312-7

Inverse cluster sets
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by T. R. Hamlett and Paul Long PDF

Proc. Amer. Math. Soc. 53 (1975), 470-476 Request permission

Abstract:

For a function $f:X \to Y$, the cluster set of $f$ at $x\epsilon X$ is the set of all $y\epsilon Y$ such that there exists a filter $\mathcal {F}$ on $X$ converging to $x$ and the filter generated by $f(\mathcal {F})$ converges to $y$. The inverse cluster set of $f$ at $y\epsilon Y$ is the set of all $x\epsilon X$ such that $y$ belongs to the cluster set of $f$ at $x$. General properties of inverse cluster sets are proved, including a necessary and sufficient condition for continuity. Necessary and sufficient conditions for functions to have a closed graph in terms of inverse cluster sets are also given. Finally, a known theorem giving a condition as to when a connected function is also a connectivity function is generalized and further investigated in terms of inverse cluster sets.

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