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

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)].

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### References:

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