Initially $\kappa$-compact spaces for large $\kappa$
Christodoulou, Stavros
Commentationes Mathematicae Universitatis Carolinae, Tome 40 (1999), p. 319-325 / Harvested from Czech Digital Mathematics Library
Access to full text
Full (PDF)

This work presents some cardinal inequalities in which appears the closed pseu\-do-\-cha\-racter, $\psi_c$, of a space. Using one of them --- $\psi_c(X) \le 2^{d(X)}$ for $T_2$ spaces --- we improve, from $T_3$ to $T_2$ spaces, the well-known result that initially $\kappa$-compact $T_3$ spaces are $\lambda $-bounded for all cardinals $\lambda $ such that $2^\lambda \leq \kappa $. And then, using an idea of A. Dow, we prove that initially $\kappa $-compact $T_2$ spaces are in fact compact for $\kappa = 2^{F(X)}$, $2^{s(X)}$, $2^{t(X)}$, $2^{\chi (X)}$, $2^{\psi_c(X)}$ or $\kappa = \max\{\tau^+, \tau^{<\tau}\}$, where $\tau > t(p,X)$ for all $p \in X$.

Publié le : 1999-01-01
Classification:  54A25,  54D10,  54D20 
@article{119087,
     author = {Stavros Christodoulou},
     title = {Initially $\kappa$-compact spaces for large $\kappa$},
     journal = {Commentationes Mathematicae Universitatis Carolinae},
     volume = {40},
     year = {1999},
     pages = {319-325},
     zbl = {0976.54022},
     mrnumber = {1732652},
     language = {en},
     url = {http://dml.mathdoc.fr/item/119087}
}

Christodoulou, Stavros. Initially $\kappa$-compact spaces for large $\kappa$. Commentationes Mathematicae Universitatis Carolinae, Tome 40 (1999) pp. 319-325. http://gdmltest.u-ga.fr/item/119087/

Balogh Z.; Dow A.; Fremlin D.H.; Nyikos P.J. Countable tightness and proper forcing, Bull. Amer. Math. Soc. 19 (1988), 295-298. (1988) | MR 0940491 | Zbl 0661.54007

Christodoulou S. Some topologies over a cardinal, Questions Answers Gen. Topology 10 (1992), 143-148. (1992) | MR 1180472 | Zbl 0782.54002

Dow A. On initially $\kappa$-compact spaces, in Rings of Continuous Functions; C. Aull ed., Lecture Notes in Pure and Applied Math. 95, 1985, pp.103-108. | MR 0789264

Dow A. Compact spaces of countable tightness in the Cohen model, in Set Theory and Its Applications; LNM, 1401, 1989, pp.55-67. | MR 1031765 | Zbl 0684.54003

Hodel R. Cardinal functions I, in Handbook of Set-Theoretic Topology; K. Kunen and J.E. Vaughan eds., North-Holland, Amsterdam, 1984, pp.1-61. | MR 0776620 | Zbl 0559.54003

Juhasz J. Cardinal functions in topology - ten years later, Mathematical Centre Tracts 123, Mathematical Centrum - Amsterdam, 1980. | MR 0576927 | Zbl 0479.54001

Juhasz J. Cardinal functions II, in Handbook of Set-Theoretic Topology; K. Kunen and J.E. Vaughan eds., North-Holland, Amsterdam, 1984, pp.63-109. | MR 0776621 | Zbl 0559.54004

Koszmider P. Splitting ultrafilters of the thin-very tall algebra and initially $ømega_1$- compactness, preprint.

Stephenson R.M., Jr. Initially $\kappa$-compact and related spaces, in Handbook Set-Theoretic Topology; K. Kunen and J.E. Vaughan eds., North-Holland, Amsterdam, 1984, pp. 603-632. | MR 0776632 | Zbl 0588.54025