The strong Pytkeev property for topological groups and topological vector spaces Academic Article

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abstract

• Generalizing the notion of metrisability, recently Tsaban and Zdomskyy (Houst J Math 35:563–571, 2009) introduced the strong Pytkeev property and proved a result stating that the space $$C_{c}(X)$$ has this property for any Polish space $$X$$. We show that the strong Pytkeev property for general topological groups is closely related to the notion of a $${\mathfrak {G}}$$-base, investigated in Gabriyelyan et al. (On topological groups with a small base and metrizability, preprint) and Ka̧kol et al. (Descriptive Topology in Selected Topics of Functional Analysis. Developments in Mathematics. Springer, Berlin, 2011). Our technique leads to an essential extension of Tsaban–Zdomskyy’s result. In particular, we prove that for a Cech-complete $$X$$ the space $$C_{c}(X)$$ has the strong Pytkeev property if and only if $$X$$ is Lindelof. We study the strong Pytkeev property for several well known classes of locally convex spaces including $$(DF)$$-spaces and strict $$(LM)$$-spaces. Strengthening results from Cascales et al. (Proc Am Math Soc 131:3623–3631, 2003) and Dudley (Proc Am Math Soc 27:531–534, 1971) we deduce that the space of distributions $${\mathfrak {D}}'(\varOmega )$$ (which is not a $$k$$-space) has the strong Pytkeev property. We also show that any topological group with a $${\mathfrak {G}}$$-base which is a $$k$$-space has already the strong Pytkeev property. We prove that, if $$X$$ is an $${\mathcal {MK}}_\omega$$-space, then the free abelian topological group $$A(X)$$ and the free locally convex space $$L(X)$$ have the strong Pytkeev property. We include various (counter) examples and pose a dozen open questions.

publication date

• January 1, 2014