Spaces not distinguishing pointwise and quasinormal convergence of real functions

The Ramsey property for C p (X)

“…First we show that there exists a perfect (non-normal) space with property QN, but without property S 1 (Γ, Γ). Recall that property wQN is preserved by continuous maps, and the closed unit interval I does not have property wQN [5].…”

“…A sequence {f n : n ∈ ω} of real-valued functions on a set X converges quasi-normally to 0 [2] if there exists a sequence {ε n : n ∈ ω} of positive real numbers converging to 0 such that for each x ∈ X, f n (x) < ε n holds for all but finitely many n ∈ ω. A space X has property QN [5] if every sequence {f n : n ∈ ω} of real-valued continuous functions on X such that f n → 0 converges quasi-normally to 0, and X has property wQN [5] if every sequence {f n : n ∈ ω} of real-valued continuous functions on X such that f n → 0 contains a subsequence which converges quasi-normally to 0.…”

On wQN∗ and wQN∗ spaces

“…A topological space X is a wQN-space if from every sequence of continuous functions converging to 0 on X one can choose a subsequence converging to 0 quasi-normally on X -see [3]. X has the sequence selection property, ✩ This research has been supported by the grant 1/3002/06 of Slovenská grantová agentúra VEGA.…”

Hurewicz sets of reals without perfect subsets

“…Theorem 2.1 is proved in three steps. The first step is analogous to Theorem 4.2 of [5] and will be used to show that the constructed set is not contained in a set of reals satisfying S 1 (C Γ , C Γ ). We say that a convergent sequence {x n } n∈N is nontrivial if lim n x n / ∈ {x n : n ∈ N}.…”