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- W2475798967 abstract "The usual ‘closed-graph theorems’ are concerned with the question of which topological structures on sets X , Y will ensure equivalence of continuity and closed-graph conditions for certain mappings t : X → Y . In this paper some stronger closed-graph like conditions are introduced on a mapping between uniform spaces, by considering the ‘hypergraph’, or graph of the induced mapping between hyperspaces. The central result, included in Theorem 1, states that for arbitrary uniform spaces X , Y , t is continuous if its hypergraph is closed. Thus for topological vector spaces (and some others) the closed-hypergraph condition is equivalent to continuity. In section 2 some situations are found in which continuity is implied by certain intermediate conditions to closed-graph and closed-hypergraph conditions. In particular, a closed-hypergraph theorem is shown to hold for locally convex spaces, provided only that X is barrelled. Finally, in section 3, the hypergraph of a relation is studied, and the separatedness of the quotient space of an equivalence relation is shown to replace continuity with regard to closed-graph and closed-hypergraph conditions." @default.
- W2475798967 created "2016-08-23" @default.
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- W2475798967 date "1974-01-01" @default.
- W2475798967 modified "2023-09-26" @default.
- W2475798967 title "9.—A Note on the Hypergraph of a Mapping between Uniform Spaces" @default.
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- W2475798967 doi "https://doi.org/10.1017/s0080454100009468" @default.
- W2475798967 hasPublicationYear "1974" @default.
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