Applications of k-covers II

Applications of bornological covering properties in metric spaces

“…x . For each x in X, Σ x denotes the collection of all sequences that converges to x [6]. Throughout the paper we assume that all the open covers of X are countably infinite.…”

Weakly infinite dimensional subsets of RN

“…S 1 (O k , Γ ) was explicitly considered in [4,Section 3], and in [5,Section 3], where the second implication of Lemma 13 is noted.…”

“…Theorem 12 of [4] shows that each σ -compact space has the property S 1 (O k , Γ ). This is not true for S 1 (O kfd , Γ ): The Hilbert cube is not weakly infinite dimensional, but is a σ -compact space.…”

“…We have m 1 < m 2 < · · · < m k , and X ⊆ k<∞ j k D n j , and each D n j is of the In Theorem 15 of [4] it was shown that a Tychonoff space X has the property S 1 (O k , Γ ) if, and only if, the space C (X) consisting of continuous real-valued functions on X has the following property: For every sequence (A n : n < ∞) of subsets of X , each having the function f ∈ A n in the compact-open topology on C (X), there is a sequence ( f n : n ∈ N) in C (X) such that for each n, f n ∈ A n , and the sequence ( f n : n < ∞) converges to f in the point-open topology of C (X). Since…”

“…(3) For each I ∈ I there is an S ⊂ X \ I such that |S| = ℵ 1 and S ∈ I. (4) For each J ∈ J there is an S ⊂ X \ J such that |S| = ℵ 1 and S ∈ J .…”