Article
Full entry | Fulltext not available
(moving wall
24 months)
Feedback
Keywords:
ring of continuous functions; $C^*$-embedded; $P$-point
ring of continuous functions; $C^*$-embedded; $P$-point
Summary:
In a Tychonoff space $X$, the point $p\in X$ is called a $C^*$-point if every real-valued continuous function on $C\smallsetminus \{p\}$ can be extended continuously to $p$. Every point in an extremally disconnected space is a $C^*$-point. A classic example is the space ${\bf W}^*=\omega_1+1$ consisting of the countable ordinals together with $\omega_1$. The point $\omega_1$ is known to be a $C^*$-point as well as a $P$-point. We supply a characterization of $C^*$-points in totally ordered spaces. The remainder of our time is aimed at studying when a point in a product space is a $C^*$-point. This process leads to many interesting new discoveries.
References:
[1] Darnel M. R.: Theory of Lattice-ordered Groups. Monographs and Textbooks in Pure and Applied Mathematics, 187, Marcel Dekker, New York, 1995. MR 1304052 | Zbl 0810.06016
[2] van Douwen E. K.: Remote Points. Dissertationes Math., Rozprawy Mat. 188 (1981), 45 pages. MR 0627526
[3] Dow A., Henriksen M., Kopperman R., Woods R. G.: Topologies and cotopologies generated by sets of functions. Houston J. Math. 19 (1993), no. 4, 551–586. MR 1251610
[4] Engelking R.: General Topology. Sigma Series in Pure Mathematics, 6, Heldermann Verlag, Berlin, 1989. MR 1039321 | Zbl 0684.54001
[5] Gillman L., Jerison M.: Rings of Continuous Functions. Graduate Texts in Mathametics, 43, Springer, New York, 1976. MR 0407579 | Zbl 0327.46040
[6] McGovern W. W.: Rings of quotients of $C(X)$ induced by points. Acta Math. Hungar. 105 (2004), no. 3, 215–230. DOI 10.1023/B:AMHU.0000049288.46182.1e | MR 2100854
Search
Partner of
