Continuous mappings on subspaces of products with the κ-box topology |
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Authors: | WW Comfort Ivan S Gotchev |
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Institution: | aDepartment of Mathematics and Computer Science, Wesleyan University, Middletown, CT 06459, USA;bDepartment of Mathematical Sciences, Central Connecticut State University, New Britain, CT 06050, USA |
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Abstract: | Much of General Topology addresses this issue: Given a function fC(Y,Z) with YY′ and ZZ′, find , or at least , such that ; sometimes Z=Z′ is demanded. In this spirit the authors prove several quite general theorems in the context Y′=(XI)κ=∏iIXi in the κ-box topology (that is, with basic open sets of the form ∏iIUi with Ui open in Xi and with Ui≠Xi for <κ-many iI). A representative sample result, extending to the κ-box topology some results of Comfort and Negrepontis, of Noble and Ulmer, and of Hušek, is this. Theorem Let ωκα (that means: κ<α, and β<α and λ<κ]βλ<α) with α regular, be a set of non-empty spaces with each d(Xi)<α, πY]=XJ for each non-empty JI such that |J|<α, and the diagonal in Z be the intersection of <α-many regular-closed subsets of Z×Z. Then (a) Y is pseudo-(α,α)-compact, (b) for every fC(Y,Z) there is JI]<α such that f(x)=f(y) whenever xJ=yJ, and (c) every such f extends to . |
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Keywords: | κ -Box topology Pseudo-(α κ )-compact κ -Invariant set Σ κ -product Calibre Souslin number Functional dependence Continuous extension of function |
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