On selectively highly divergent spaces
Keywords:
sequences, compactifications, remainder, realcompact, absolutes, F-spaces, isolated pointsAbstract
In this paper we introduce the class of SHD spaces, defined by a selection property, and study their fundamental topological features. We exhibit that such class is full of variety: in particular, we construct an SHD space that simultaneously contains a convergent sequence and a dense subspace where the only convergent sequences are the eventually constant ones. We prove that if $X$ is regular and for all $x \in X$ holds $\psi(x, X)>\omega$, then $X_\delta$ is SHD. Finally, for any Hausdorff space $X$ without isolated points, we produce an associated space $s X$ which is extremely disconnected, zero-dimensional, Hausdorff and SHD, and which satisfies $|X|=|s X|, \pi w(X)=\pi w(s X)$ and $c(X)=c(s X)$.
References
A. Dorantes-Aldama and D. Shakhmatov, Compactness properties defined by open-point games, Topology and its Applications 284 (2019), 1-21.
A. V. Arhangel'skii, $G_\delta$-modification of compacta and cardinal invariants, Commentationes Mathematicae Universitatis Carolinae 47 (2006), no. 1, 95-101.
C. E. Aull, Sequences in topological spaces, Annales societatis mathematicae polonae (1968), 329-336.
D. J. Lutzer, Pixley-Roy Topology, Topology Proceedings 3 (1978), 139-158.
F. Galvin, Indeterminacy of point-open games, Bull. Acad. Pol. Sci., Sér. Sci. Math. Astron. Phys. 26 (1978), 445-449.
F. Rothberger, Eine Verschärfung der Eigenschaft $C$, Fundamenta Mathematicae 30 (1938), no. 1, 50-55.
G. B. Sorin, The Connectedness of the Stone-Crech Remainder, Moscow University Mathematics Bulletin 75 (2020), no. 2, 47-49.
I. Juhász, Cardinal Functions in Topology, Ten Years Later, Math. Centre Tracts, Michigan, 1980.
J. Porter and R. Woods, Extensions and Absolutes of Hausdorff Spaces, SpringerVerlag, New York, 1988.
K. Kunen and J. E. Vaughan (eds.), Handbook of set-theoretic topology, Elsevier Science Publishers, New York, 1984.
K. Menger, Selecta Mathematica I (B. Schweizer, A. Sklar, K. Sigmund, P. Gruber, E. Hlawka, L. Reich, and L. Schmetterer, eds.), Spinger Vienna, 2002.
L. Gillman and M. Jerison, Rings of Continuous Functions, Van Nostrand, New York, 1960.
M. Hrušák, A. Tamariz-Mascarúa, and M. Tkachenko (eds.), Pseudocompact Topological Spaces: A Survey of Classic and New Results with Open Problems, Vol. 55, Springer, Switzerland.
M. Sakai, Cardinal Functions of Pixley-Roy hyperspaces, Topology and its Applications 159 (2012), 3080-3088.
M. Scheepers, Combinatorics of open covers I: Ramsey theory, Topology and its Applications 69 (1996), no. 1, 31-62.
R. Engelking, General Topology: Revised and completed edition, Heldermann, Berlin, 1989.
R. Hernández-Gutiérrez and A. Tamariz-Mascarúa, Disconnectedness properties of hyperspaces, Commentationes Mathematicae Universitatis Carolinae 52 (2011), no. 4, 569-591.
R. C. Walker, The Stone-Crech compactification, Springer-Verlag, Pittsburgh, 1974.
R. Telgársky, Spaces defined by topological games, Fundamenta Mathematicae 88 (1975), no. 3, 193-223.
T. Isiwata, Some properties of the remainder of Stone-Crech compactifications, Fundamenta Mathematicae 83 (1974), no. 2, 129-142.
V. V. Tkachuk, A $C_p$-Theory Problem Book: Topological and Function Spaces, Springer-Verlag, New-York, 2011.
W. Hurewicz, Über eine Verallgemeinerung des Borelschen Theorems, Mathematische Zeitschrift 24 (1926), 401-421.
W. W. Comfort and S. Negrepontis, Chain Conditions in Topology, Cambridge University Press, Cambridge, 1982.
