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- W1036128583 abstract "This chapter considers two topological spaces X and Y to illustrate the idea of product spaces. The product topology on X × Y is the topology having the collection B = {U ×V : U is open in X, V is open in Y} as a base. A product set X × Y with the product topology is called the product (or product space) of X and Y. If R is the real line with the usual topology, then the product topology on R2 (equal to R×R) coincides with the usual topology of the plane R2. Normality, is not preserved by product spaces. For example, if the Sorgenfrey line S is the set R of all real numbers, having the set {(a, b): a < b} as a base, then S is normal and S2, which is equal to S × S, is not normal. The Sorgenfrey line S is a regular Lindelöf space. Because every regular Lindelöf space is a paracompact Hausdorff space and every paracompact Hausdorff space is normal, neither Lindelöfness nor paracompactness is preserved by product spaces in general. However, if X is compact and Y is paracompact, then X × Y is paracompact." @default.
- W1036128583 created "2016-06-24" @default.
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- W1036128583 date "2003-01-01" @default.
- W1036128583 modified "2023-09-25" @default.
- W1036128583 title "Product Spaces" @default.
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- W1036128583 doi "https://doi.org/10.1016/b978-044450355-8/50011-8" @default.
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