Discrete and disjoint shrinking properties of locally convex spaces. (English) Zbl 1538.54062

Tkachuk, V. V.

Acta Math. Hung. 170, No. 2, 709-720 (2023).

Selection principles represent a very popular trend in modern topology, see e.g. G. Gruenhage and M. Sakai [Topology Appl. 158, No. 12, 1352–1359 (2011; Zbl 1228.54028)], or L. D. R. Kočinac [Quaest. Math. 43, No. 8, 1121–1153 (2020; Zbl 1450.54010)]. Three kinds of such properties are considered in the paper under review. A topological space \(X\) has the disjoint shrinking property (discrete shrinking property) if, for every sequence \(\{ U_n\colon n\in\omega\}\) of non-empty open subsets of \(X\), there exists a disjoint (respectively, discrete) family \(\{ V_n\colon n\in\omega\}\) of non-empty open sets such that \(V_n\subset U_n\) for each \(n\in\omega\). These notions were introduced and sudied by V. V. Tkachuk in [J. Math. Anal. Appl. 508, No. 1, Article ID 125865, 12 p. (2022; Zbl 1481.54020)]. In the paper under review, the author strengthens some results and answers several open questions from the previous article.
The disjoint shrinking property is considered in the first part of the paper. The key observation here is that a non-empty Tychonoff topological space \(X\) has the disjoint shrinking property if and only if any non-empty open subspace \(U\) of \(X\) has uncountable \(\pi\)-weight. It follows that a topological group \(G\) has the disjoint shrinking property if and only if every non-empty open subset of \(G\) has uncountable weight, and a locally convex space has the disjoint shrinking property if and only if its weight is uncountable. In particular, any non-metrizable locally convex space has the disjoint shrinking property.
The discrete shrinking property is studied in the second part of the paper. It is shown that a locally convex space \(L\) has the discrete shrinking property if and only if it is not metrizable.
Finally, the author proves that if \(X\) is a space with a unique non-isolated point \(p\), then the space \(C_p(X, [0,1])\) is discretely selective if and only if \(X\setminus A\) is not a \(P\)-space for any countable set \(A\subset X\setminus\{p\}\). Recall that a space \(X\) is discretely selective if for any family \(\{ U_n\colon n\in\omega\}\) of non-empty open subsets of \(X\) there exists a selection \(\{ x_n\colon n\in\omega\}\) which is closed and discrete in \(X\), see V. V. Tkachuk [Acta Math. Hung. 154, No. 1, 56–68 (2018; Zbl 1399.54047)].

Reviewer: Tomasz Natkaniec (Gdańsk)

MSC:

54C35	Function spaces in general topology
54C05	Continuous maps
54G20	Counterexamples in general topology

Keywords:

selection; function space; locally convex space; topological group; discretely selective space; \(\pi\)-character; closed discrete set; discrete shrinking property; disjoint shrinking property; \(P\)-space; essentially uncountable space

Citations:

Zbl 1228.54028; Zbl 1450.54010; Zbl 1481.54020; Zbl 1399.54047

References:

[1]	A. V. Arhangel’skii, Relations among the invariants of topological groups and their subspaces, Russian Math. Surveys, 35 (1980), 1-23. · Zbl 0458.22002
[2]	L. F. Aurichi and R. R. Dias, Topological games and Alster spaces, Canadian Math. Bull., 57 (2014), 683-696. · Zbl 1306.54029
[3]	L. Babinkostova, On some questions about selective separability, Math. Log. Quart., 55 (2009), 260-262.
[4]	L. Babinkostova, L. D. R. Kočinac and M. Scheepers, Combinatorics of open covers (VIII), Topology Appl., 140 (2004), 15-32. · Zbl 1051.54019
[5]	A. Bella, M. Bonanzinga, M. V. Matveev and V. V. Tkachuk, Selective separability: general facts and behavior in countable spaces, Topology Proc., 32 (2008), 15- 30. · Zbl 1165.54008
[6]	R. Engelking, General Topology, PWN (Warszawa, 1977). · Zbl 0373.54002
[7]	G. Gruenhage and M. Sakai, Selective separability and its variations, Topology Appl., 158 (2011), 1352-1359. · Zbl 1228.54028
[8]	D. Guerrero Sánchez and V. V. Tkachuk, If C_p(X) is strongly dominated by a second countable space, then X is countable, J. Math. Anal. Appl., 454 (2017), 533- 541. · Zbl 1394.54001
[9]	R.E. Hodel, Cardinal functions. I, in: Handbook of Set-Theoretic Topology, ed. by K. Kunen and J. E. Vaughan, North Holland (Amsterdam, 1984), pp. 1-61. · Zbl 0559.54003
[10]	M. Scheepers, Combinatorics of open covers (I): Ramsey theory, Topology Appl., 69 (1996), 31-62. · Zbl 0848.54018
[11]	M. Scheepers, Combinatorics of open covers (III): games, C_p(X), Fund. Math., 152 (1997), 231-254. · Zbl 0884.90149
[12]	M. Scheepers, Combinatorics of open covers (VI): selectors for sequences of dense sets, Quaestiones Math., 22 (1999), 109-130. · Zbl 0972.91026
[13]	M. Scheepers, Selection principles and covering properties in Topology, Note Mat., 22 (2003), 3-41. · Zbl 1195.37029
[14]	V. V. Tkachuk, A C_p-Theory Problem Book. Topological and Function Spaces, Springer (New York, 2011). · Zbl 1222.54002
[15]	V. V. Tkachuk, A C_p-Theory Problem Book. Special Features of Function Spaces, Springer (New York, 2014). · Zbl 1303.54006
[16]	V. V. Tkachuk, A C_p-Theory Problem Book. Compactness in Function Spaces, Springer (New York, 2015). · Zbl 1325.54001
[17]	V. V. Tkachuk, Strong domination by countable and second countable spaces, Topology Appl., 228 (2017), 318-326. · Zbl 1381.54018
[18]	V. V. Tkachuk, Closed discrete selections for sequences of open sets in function spaces, Acta Math. Hungar., 154 (2018), 56-68. · Zbl 1399.54047
[19]	V. V. Tkachuk, Open discrete shrinkings in function spaces, J. Math. Anal. Appl., 508 (2022), Paper No. 125865, 12 pp. · Zbl 1481.54020