Continuous selections and σ-spaces

0 M ay 2 00 7 CONTINUOUS SELECTIONS AND σ - SPACES

Assume that X is a metrizable separable space, and each clopen-valued lower semicontinuous multivalued map Φ : X ⇒ Q has a continuous selection φ : X → Q. Our main result is that in this case, X is a σ-space. We also derive a partial converse implication, and present a reformulation of the Scheepers Conjecture in the language of continuous selections.

Continuous Selections and Approximations in Α-convex Metric Spaces

In the paper, the notion of a generalized convexity was defined and studied from the view-point of the selection and approximation theory of set-valued maps. We study the simultaneous existence of continuous relative selections and graph-approximations of lower semicontinuous and upper semicontinuous set-valued maps with α-convex values having nonempty intersection.

σ-Normality of Topological Spaces

Selections on Ψ - spaces

We show that if A is an uncountable AD (almost disjoint) family of subsets of ω then the space Ψ(A) does not admit a continuous selection; moreover, if A is maximal then Ψ(A) does not even admit a continuous selection on pairs, answering thus questions of T. Nogura.

Selections and Finite C - Spaces

Characterizations of paracompact finite C-spaces via continuous selections avoiding Zσ-sets are given. We apply these results to obtain some properties of finite C-spaces. Factorization theorems and a completion theorem for finite C-spaces are also proved.