Inverse cluster sets
| Autor: | T. R. Hamlett, Paul E. Long |
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| Rok vydání: | 1975 |
| Předmět: | |
| Zdroj: | Proceedings of the American Mathematical Society. 53:470-476 |
| ISSN: | 1088-6826 0002-9939 |
| DOI: | 10.1090/s0002-9939-1975-0388312-7 |
| Popis: | For a function f : X → Y f:X \to Y , the cluster set of f f at x ϵ X x\epsilon X is the set of all y ϵ Y y\epsilon Y such that there exists a filter F \mathcal {F} on X X converging to x x and the filter generated by f ( F ) f(\mathcal {F}) converges to y y . The inverse cluster set of f f at y ϵ Y y\epsilon Y is the set of all x ϵ X x\epsilon X such that y y belongs to the cluster set of f f at x 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. |
| Databáze: | OpenAIRE |
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