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Transactions of the American Mathematical Society

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Ultrafilter invariants in topological spaces


Author: Victor Saks
Journal: Trans. Amer. Math. Soc. 241 (1978), 79-97
MSC: Primary 54A20; Secondary 54A25, 54D20
DOI: https://doi.org/10.1090/S0002-9947-1978-0492291-9
MathSciNet review: 492291
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Abstract: Let $ \mathfrak{m}\, \geqslant \,{\aleph _0}$ and $ X\, = \,\prod\nolimits_{i\, \in \,I} {{X_i}} $. Then X is $ [{\aleph _0},\,\mathfrak{m}]$-compact if and only if $ \prod\nolimits_{i\, \in \,J} {{X_i}} $ is $ [{\aleph _0},\,\mathfrak{m}]$-compact for all $ J\, \subset \,I$ with $ \vert J\vert\, \leqslant \,{2^{{2^\mathfrak{m}}}}$. Let $ \mathfrak{m}\, \geq \,{\aleph _0}$, $ ({x_\xi }:\,\xi \, < \,\mathfrak{m})$ a net in X, $ p\, \in \,X$, and $ \mathcal{D}\, \in \,\beta (\mathfrak{m})$. Then $ p\, = \,\mathcal{D}\, - \,{\lim _{\xi < \mathfrak{m}}}\,{x_\xi }$ if $ \{ \xi \, < \,\mathfrak{m}:\,{x_\xi }\, \in \,U \} \, \in \, \mathcal{D}$ for every neighborhood U of p. Every topological space is characterized by its $ \mathcal{D}$-limits. Several topological properties are described using ultrafilter invariants, including compactness and perfect maps. If X is a Hausdorff space and D is a discrete space equipotent with a dense subset of X, then X is a continuous perfect image of a subspace of $ \beta D$ which contains D if and only if X is regular.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1978-0492291-9
Keywords: Ultrafilter, $ \mathcal{D}$-limit point, $ \mathcal{D}$-compact, countably compact product, $ [\mathfrak{n},\,\mathfrak{m}]$-compact, complete accumulation point, $ C[\mathfrak{n},\,\mathfrak{m}]$, perfect map, regular space
Article copyright: © Copyright 1978 American Mathematical Society

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