Topological vector spaces and function spaces which are $$\kappa $$-Fréchet–Urysohn and have open countable tightness

Abstract We characterize topological vector spaces (tvs) which are $$\kappa $$ κ -Fréchet–Urysohn or have open countable tightness. It is shown that an infinite-dimensional feral locally convex space is not $$\kappa $$ κ -Fréchet–Urysohn. It is proved that the product of a metrizable tvs and a tvs with open countable tightness also has open countable tightness. Let Y be a non-trivial metrizable abelian group, $$\mathcal {I}$$ I be an ideal of compact subsets of a topological space X . Denote by $$C_\mathcal {I}(X,Y)$$ C I ( X , Y ) and by $$C_\mathcal {I}^*(X,Y)$$ C I ∗ ( X , Y ) the space C ( X , Y ) of all continuous functions and the space $$C^*(X,Y)$$ C ∗ ( X , Y ) of all continuous functions with relatively compact image endowed with the topology defined by the ideal $$\mathcal {I}$$ I . Generalizing results of M. Sakai and the author it is shown that if X is a strongly (resp., $$\text {strongly}^*$$ strongly ∗ ) $$Y_\mathcal {I}$$ Y I -Tychonoff space, then (1) $$C_\mathcal {I}(X,Y)$$ C I ( X , Y ) (resp., $$C_\mathcal {I}^*(X,Y)$$ C I ∗ ( X , Y ) ) has open countable tightness iff every $$\mathcal {I}$$ I -moving off family $$\mathcal {K}\subseteq \mathcal {I}$$ K ⊆</