Wallman-type compactifications and products
HTML articles powered by AMS MathViewer
- by Frank Kost PDF
- Proc. Amer. Math. Soc. 29 (1971), 607-612 Request permission
Abstract:
Y is a Wallman-type compactification (O. Frink, Amer. J. Math. 86 (1964), 602-607) of X in case there is a normal base Z for the closed sets of X such that the ultrafilter space from Z, denoted $\omega (Z)$, is topologically Y. It is not known if every compactification is Wallman-type. For ${Z_\alpha }$ a normal base for the closed sets of ${X_\alpha }$ for each a belonging to an index set $\Delta$ it is shown that the Tychonoff product space ${\prod _{\alpha \in \Delta }}\omega ({Z_\alpha })$ is a Wallman compactification of ${\prod _{\alpha \in \Delta }}{X_\alpha }$. Also for $X \subset T \subset \omega (Z)$ with Z a normal base for the closed sets of X, a proof that $\omega (Z)$ is a Wallman-type compactification of T is indicated.References
- R. M. Brooks, On Wallman compactifications, Fund. Math. 60 (1967), 157–173. MR 210069, DOI 10.4064/fm-60-2-157-173
- Orrin Frink, Compactifications and semi-normal spaces, Amer. J. Math. 86 (1964), 602–607. MR 166755, DOI 10.2307/2373025
- Leonard Gillman and Meyer Jerison, Rings of continuous functions, The University Series in Higher Mathematics, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto-London-New York, 1960. MR 0116199
- E. F. Steiner, Wallman spaces and compactifications, Fund. Math. 61 (1967/68), 295–304. MR 222849, DOI 10.4064/fm-61-3-295-304
- Henry Wallman, Lattices and topological spaces, Ann. of Math. (2) 39 (1938), no. 1, 112–126. MR 1503392, DOI 10.2307/1968717
Additional Information
- © Copyright 1971 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 29 (1971), 607-612
- MSC: Primary 54.53
- DOI: https://doi.org/10.1090/S0002-9939-1971-0281159-8
- MathSciNet review: 0281159