Some kinds of perfect spaces, including paracompact perfectly normal spaces and collectionwise normal perfect spaces, are characterized in terms of continuous selections avoiding supporting sets. A necessary and sufficient condition on a domain space for a selection theorem of E. Michael [Fund. Math. 47 (1959), 173-178] to hold is also obtained.

A classical Lefschetz result about point-finite open covers of normal spaces is generalised by showing that every lower semi-continuous mapping from a normal space into the nonempty compact subsets of a metrizable space admits a closed graph multi-selection. Several applications are given as well.

We present a new continuous selection theorem, which unifies in some sense two well known selection theorems; namely we prove that if F is an H-upper semicontinuous multivalued map on a separable metric space X, G is a lower semicontinuous multivalued map on X, both F and G take nonconvex $L_p(T,E)$-decomposable closed values, the measure space T with a σ-finite measure μ is nonatomic, 1 ≤ p < ∞, $L_p(T,E)$ is the Bochner-Lebesgue space of functions defined on T with values in a Banach space E, F(x) ∩ G(x) ≠ ∅...

We use Ramseyan partition relations to characterize: ∙ the classical covering property of Hurewicz; ∙ the covering property of Gerlits and Nagy; ∙ the combinatorial cardinal numbers and add(ℳ ). Let X be a $T_{31/2}$-space. In [9] we showed that $C_p\left(X\right)$ has countable strong fan tightness as well as the Reznichenko property if, and only if, all finite powers of X have the Gerlits-Nagy covering property. Now we show that the following are equivalent: 1. $C_p\left(X\right)$ has countable fan tightness and the Reznichenko property. 2....

We consider a concave iteration semigroup of linear continuous set-valued functions defined on a closed convex cone in a separable Banach space. We prove that such an iteration semigroup has a selection which is also an iteration semigroup of linear continuous functions. Moreover it is majorized by an "exponential" family of linear continuous set-valued functions.

It is shown that every metrizable consonant space is a Cantor set-selector. Some applications are derived from this fact, also the relationship is discussed in the framework of hyperspaces and Prohorov spaces.

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.

Every l.s.cṁapping from a paracompact space into the non-empty, closed, convex subsets of a (not necessarily convex) $G_{\delta }$-subset of a Banach space admits a single-valued continuous selection provided every such mapping admits a convex-valued usco selection. This leads us to some new partial solutions of a problem raised by E. Michael.

The existence of continuous selections is proved for a class of lower semicontinuous multifunctions whose values are closed convex subsets of a complete metric space equipped with an appropriate notion of convexity. The approach is based on the notion of pseudo-barycenter of an ordered n-tuple of points.

For a space $Z$, we denote by $ℱ\left(Z\right)$, $𝒦\left(Z\right)$ and ${ℱ}_2\left(Z\right)$ the hyperspaces of non-empty closed, compact, and subsets of cardinality $\le 2$ of $Z$, respectively, with their Vietoris topology. For spaces $X$ and $E$, $C_p(X,E)$ is the space of continuous functions from $X$ to $E$ with its pointwise convergence topology. We analyze in this article when $ℱ\left(Z\right)$, $𝒦\left(Z\right)$ and ${ℱ}_2\left(Z\right)$ have continuous selections for a space $Z$ of the form $C_p(X,E)$, where $X$ is zero-dimensional and $E$ is a strongly zero-dimensional metrizable space. We prove that $C_p(X,E)$ is weakly orderable if and...

Let E be a locally convex topological Hausdorff space, K a nonempty compact convex subset of E, μ a regular Borel probability measure on E and γ > 0. We say that the measure μ γ-represents a point x ∈ K if $su{p}_{\left|\right|f\left|\right|\le 1}|f\left(x\right)-{\int }_{K}fd\mu |<\gamma$ for any f ∈ E*. In this paper a continuous version of the Choquet theorem is proved, namely, if P is a continuous multivalued mapping from a metric space T into the space of nonempty, bounded convex subsets of a Banach space X, then there exists a weak* continuous family $\left({\mu }_t\right)$ of regular Borel...