B. J. Day and G. M. Kelly, On topological quotients preserved by pullback or products, Proc. Camb. Phil. Soc. 67 (1970) 553--558. Home/Search Document Not in Database Summary Related Articles Check |
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Exponentiability Of Perfect Maps: Four Approaches - Niefield (2002) (Correct)
....clearly stated, and it was shown that a separable metric space is exponentiable if and only if it is locally compact. Fox attributed his interest in this problem to a question posed to him in a letter from Hurewicz. A complete characterization of exponentiable spaces appeared in the 1970 paper [DK] of Day and Kelly where they proved that the functor Y preserves quotient maps if and only if the lattice ) of open subsets of Y is a continuous lattice in the sense of Scott [S] and that a Hausdor# space satisfies this property if and only if it is locally compact. Since Y preserves ....
....subspace. More recently, Lowen Colebunders and Richter [LR] showed that the exponentials Z in Top can be described using the way below relation on the continuous lattice ) and then Richter [R] generalized this result to the fiberwise case. For a further discussion of the influence of [DK], the reader is referred to Isbell s article General function spaces, products, and continuous lattices [I] For more on exponentiability, see also [N4] 3. Perfect Maps Recall that a continuous function p: Y T is called proper if it closed and has compact fibers, separated if the diagonal ....
B. J. Day and G. M. Kelly, On topological quotients preserved by pullback or products, Proc. Camb. Phil. Soc. 67 (1970) 553--558.
Exponentiable Morphisms: Posets, Spaces, Locales, And.. - Niefield (Correct)
....condition of local compactness. and showed that a separable metrizable space is exponentiable if and only if it is locally compact. The characterization of exponentiable spaces was finally achieved by Day and Kelly in their 1970 paper On topological quotients preserved by pullback or products [DK], where they proved that, for a space X, the functor Gamma Theta X:Top Gamma Top Theory and Applications of Categories, Vol. 8, No. 2 18 preserves quotients if and only if the lattice O(X) of open sets of X is (what is now known as) a continuous lattice, in the sense of Scott [S] i.e. O(X) ....
....only if it is a continuous category. In addition, they showed that for a locale X, the topos Sh(X) of sheaves on X is exponentiable in GTop if and only if X is metastably locally compact. This is a property implying local compactness but using a strengthening of in its place. The results of [DK] were also relativized by Niefield in [N3] where the characterization of exponentiable spaces was generalized to Top=T . As corollaries, it was shown that if X is locally compact and T is Hausdorff, then any continuous map X Gamma T is exponentiable in Top, and if X is a subspace of T , then ....
B. J. Day and G. M. Kelly, On topological quotients preserved by pullback or products, Proc. Camb. Phil. Soc. 67, 1970, 553--558.
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