A characterization of a space with countable infinity
HTML articles powered by AMS MathViewer
- by Akihiro Okuyama PDF
- Proc. Amer. Math. Soc. 28 (1971), 595-597 Request permission
Abstract:
It is well known that, for a countable discrete space $N$, \[ |\beta N - N| = {2^{{2^{{\aleph _0}}}}}.\] So, any completely regular ${T_1}$ space $X$ with $\beta X - X| \leqq {\aleph _0}$ does not contain any infinite discrete subspace. In this paper, we characterize those completely regular ${T_1}$ spaces with countable infinity as follows: Such a space $X$ is characterized by the two properties. (a) $X$ is pseudocompact. (b) There exist a compact metric space $Y$ and a continuous map $f$ from $X$ onto $Y$ so that the subset ${Y_0} = \{ y:{f^{ - 1}}(y)$ of $Y$ is countable and $c{l_{\beta X}}{f^{ - 1}}(y) - {f^{ - 1}}(y)$ is one point whenever $y \in {Y_0}$. (In particular, for any $y$ in ${Y_0},\beta ({f^{ - 1}}(y)) - {f^{ - 1}}(y)$ is one point if $X$ is normal.)References
- 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
- Takesi Isiwata, Mappings and spaces, Pacific J. Math. 20 (1967), 455–480; correction, ibid. 23 (1967), 630–631. MR 0219044, DOI 10.2140/pjm.1967.23.630
- K. D. Magill Jr., $N$-point compactifications, Amer. Math. Monthly 72 (1965), 1075–1081. MR 185572, DOI 10.2307/2315952
- Kenneth D. Magill Jr., Countable compactifications, Canadian J. Math. 18 (1966), 616–620. MR 198420, DOI 10.4153/CJM-1966-060-6
Additional Information
- © Copyright 1971 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 28 (1971), 595-597
- MSC: Primary 54.53
- DOI: https://doi.org/10.1090/S0002-9939-1971-0276929-6
- MathSciNet review: 0276929