Tychonoff expansions with prescribed resolvability properties |
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Authors: | WW Comfort |
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Institution: | a Department of Mathematics and Computer Science, Wesleyan University, Wesleyan Station, Middletown, CT 06459, United States b Department of Mathematics and Computer Science, Albany State University, Albany, GA 31705, United States |
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Abstract: | The recent literature offers examples, specific and hand-crafted, of Tychonoff spaces (in ZFC) which respond negatively to these questions, due respectively to Ceder and Pearson (1967) 3] and to Comfort and García-Ferreira (2001) 5]: (1) Is every ω-resolvable space maximally resolvable? (2) Is every maximally resolvable space extraresolvable? Now using the method of KID expansion, the authors show that every suitably restricted Tychonoff topological space (X,T) admits a larger Tychonoff topology (that is, an “expansion”) witnessing such failure. Specifically the authors show in ZFC that if (X,T) is a maximally resolvable Tychonoff space with S(X,T)?Δ(X,T)=κ, then (X,T) has Tychonoff expansions U=Ui (1?i?5), with Δ(X,Ui)=Δ(X,T) and S(X,Ui)?Δ(X,Ui), such that (X,Ui) is: (i=1) ω-resolvable but not maximally resolvable; (i=2) if κ′ is regular, with S(X,T)?κ′?κ] τ-resolvable for all τ<κ′, but not κ′-resolvable; (i=3) maximally resolvable, but not extraresolvable; (i=4) extraresolvable, but not maximally resolvable; (i=5) maximally resolvable and extraresolvable, but not strongly extraresolvable. |
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Keywords: | primary 05A18 03E05 54A10 secondary 03E35 54A25 05D05 |
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