We study topological properties of Valdivia compact spaces. We prove in particular that a compact Hausdorff space K is Corson provided each continuous image of K is a Valdivia compactum. This answers a question of M. Valdivia (1997). We also prove that the class of Valdivia compacta is stable with respect to arbitrary products and we give a generalization of the fact that Corson compacta are angelic.
It is shown that there exists a -compact topological group which cannot be represented as a continuous image of a Lindelöf -group, see Example 2.8. This result is based on an inequality for the cardinality of continuous images of Lindelöf -groups (Theorem 2.1). A closely related result is Corollary 4.4: if a space is a continuous image of a Lindelöf -group, then there exists a covering of by dyadic compacta such that . We also show that if a homogeneous compact space is a continuous...
A compact metric space X̃ is said to be a continuous pseudo-hairy space over a compact space X ⊂ X̃ provided there exists an open, monotone retraction such that all fibers are pseudo-arcs and any continuum in X̃ joining two different fibers of r intersects X. A continuum is called a continuous pseudo-fan of a compactum X if there are a point and a family ℱ of pseudo-arcs such that , any subcontinuum of intersecting two different elements of ℱ contains c, and ℱ is homeomorphic to X (with...
Continuous approximation selection theorems are given. Hence, in some special cases continuous versions of Fillipov's selection theorem follow.
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.