Wallman-type compactifications
Author:
Charles M. Biles
Journal:
Proc. Amer. Math. Soc. 25 (1970), 363-368
MSC:
Primary 54.53
DOI:
https://doi.org/10.1090/S0002-9939-1970-0263029-3
MathSciNet review:
0263029
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Abstract | References | Similar Articles | Additional Information
Abstract: All spaces in this paper are Tychonoff. A Wallman base on a space $X$ is a normal separating ring of closed subsets of $X$ (see Steiner, Duke Math. J. 35 (1968), 269-276). Let $T$ be a compact space and $\mathcal {L}$ a Wallman base on $T$. For $X \subset T$, define ${\mathcal {L}_X} = \{ A \cap X|A \in \mathcal {L}\}$. Theorem 1. If $X$ is a dense subspace of $T$, then $T = w{\mathcal {L}_X}$ iff ${\operatorname {cl} _T}A \cap {\operatorname {cl} _T}B = \emptyset$ whenever $A,B \in {\mathcal {L}_X}$ and $A \cap B = \emptyset$. Theorem 2. $T = w{\mathcal {L}_X}$ for each dense $X \subset T$ iff $T = w{\mathcal {L}_Y}$ for each dense $Y \subset T$ where $T - Y \in \mathcal {L}$. From these theorems we show that every compact $F$-space and every compact orderable space is a Wallman compactification of each of its dense subspaces.
- Richard A. AlΓ² and Harvey L. Shapiro, Normal bases and compactifications, Math. Ann. 175 (1968), 337β340. MR 220246, DOI https://doi.org/10.1007/BF02063218
- R. M. Brooks, On Wallman compactifications, Fund. Math. 60 (1967), 157β173. MR 210069, DOI https://doi.org/10.4064/fm-60-2-157-173
- Orrin Frink, Compactifications and semi-normal spaces, Amer. J. Math. 86 (1964), 602β607. MR 166755, DOI https://doi.org/10.2307/2373025
- Anthony W. Hager, On inverse-closed subalgebras of $C(X)$, Proc. London Math. Soc. (3) 19 (1969), 233β257. MR 244948, DOI https://doi.org/10.1112/plms/s3-19.2.233
- Anthony W. Hager and Donald G. Johnson, A note on certain subalgebras of $C({\mathfrak X})$, Canadian J. Math. 20 (1968), 389β393. MR 222647, DOI https://doi.org/10.4153/CJM-1968-035-4
- R. Kaufman, Ordered sets and compact spaces, Colloq. Math. 17 (1967), 35β39. MR 212769, DOI https://doi.org/10.4064/cm-17-1-35-39
- John L. Kelley, General topology, D. Van Nostrand Company, Inc., Toronto-New York-London, 1955. MR 0070144
- E. F. Steiner, Wallman spaces and compactifications, Fund. Math. 61 (1967/68), 295β304. MR 222849, DOI https://doi.org/10.4064/fm-61-3-295-304
- A. K. Steiner and E. F. Steiner, Wallman and $Z$-compactifications, Duke Math. J. 35 (1968), 269β275. MR 227942
- 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
- Henry Wallman, Lattices and topological spaces, Ann. of Math. (2) 39 (1938), no. 1, 112β126. MR 1503392, DOI https://doi.org/10.2307/1968717
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Additional Information
Keywords:
Tychonoff space,
compact space,
Hausdorff compactification,
Wallman base,
Wallman compactification,
<IMG WIDTH="16" HEIGHT="20" ALIGN="BOTTOM" BORDER="0" SRC="images/img5.gif" ALT="$z$">-compactification,
dense subspace,
compact <IMG WIDTH="21" HEIGHT="20" ALIGN="BOTTOM" BORDER="0" SRC="images/img1.gif" ALT="$F$">-space,
compact orderable space
Article copyright:
© Copyright 1970
American Mathematical Society