Spaces with a property related to uniformly local finiteness

Extensions by Means of Expansions and Selections

“…Next, construct a strictly increasing sequence m n : n 2 f Ng & N such that f n (p)(m n ) < 1/n for every n 2 N: Now, for every 2 R, with For an infinite cardinal !, a space X is said to have the property U ! À Á if every locally finite collection F of subsets of X, with F j j r !, is uniformly locally finite, see [21]. Also, let us recall that a map g: X !…”

“…As it was shown in [21], a space X has the property U ! ð Þ if and only if X is a cb-space in the sense of Mack [24].…”

Does C* -embedding imply C*-embedding in the realm of products with a non-discrete metric factor?

“…In [8] Hoshina has defined a subset A of X to be U λ -embedded in X if every uniformly locally finite (in A) family F of subsets of A with |F| ≤ λ is uniformly locally finite in X. The equivalence stated by Lemma 2.5 justifies the use of the same name for the embedding property in the Introduction defined as "U ω -embedding".…”

“…The notion "U ω -embedded" in this sense is the same as "U ω -embedded" in the sense of Hoshina [8] (see Lemma 2.5). It should be mentioned that a subset A ⊂ X is C-embedded in X if and only if it is both U ω -and C * -embedded in X [20] (see [8,Proposition 1.6]).…”

“…Now, we shall say that a subset A ⊂ X is U ω -embedded in X if for every continuous function f : A → R, there exists a continuous function g : X → R such that f (x) ≤ g(x) whenever x ∈ A. The notion "U ω -embedded" in this sense is the same as "U ω -embedded" in the sense of Hoshina [8] (see Lemma 2.5). It should be mentioned that a subset A ⊂ X is C-embedded in X if and only if it is both U ω -and C * -embedded in X [20] (see [8,Proposition 1.6]).…”