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Many Eberlein-Grothendieck spaces have no non-trivial convergent sequences. (English) Zbl 1406.54010

Let \(X\) be a Tychonoff space and \(C_p(X)\) the set of all real-valued continuous functions on \(X\) equipped with the pointwise convergence topology. In the paper under review, the author proves the following results: (1) if \(X\) is a \(\sigma\)-pseudocompact space, then every \(k\)-subspace of \(C_p(X)\) has the Fréchet-Urysohn property. (2) If \(X\) is a countably compact sequential space and \(C_p(X)\) has a dense \(k\)-subspace, then \(X\) is scattered. (3) if \(X\) is a second countable uncountable \(\sigma\)-space, then any dense subset of \(C_p(X)\) has a dense subspace without non-trivial convergent sequences. (4) It is also proved that if \(X\) is a monolithic compact space, then \(X\) is scattered if and only if every dense subset of \(C_p(X)\) has a non-trivial convergent sequence if and only if \(C_p(X)\) is Fréchet-Urysohn. As a consequence, for any cardinal \(\kappa \geq c\), the space \(\mathbb{R}^{\kappa}\) has a dense subset with no non-trivial convergent sequences.

MSC:

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