arXiv subjects

Ľubica Holá

Publications and source records attributed to Ľubica Holá.

5 recordsLinked to original sources

arXiv2026

Baire 1 functions and the strong Choquet property

Let X be a Tychonoff space and B_1(X) be the space of real-valued Baire 1 functions. Using the classical Lebesgue's characterization of Baire 1 functions and the strong Choquet game we prove that if X is locally compact or a topological sum of sigma-compact spaces, the space B_1(X) equipped with the topology τ_UC of uniform convergence on compacta, is a strong Choquet space. For the Sorgenfrey line S, the space (B_1(S); τ_UC) is of the first Baire category in itself.

math.GN↗

arXiv2026

Completeness properties of the space of quasicontinuous functions

Quasicontinuous functions have found applications in many areas of mathematics. We study completeness properties of the space of quasicontinuous functions equipped with the topology of pointwise convergence. Let X be a Hausdorff topological space, Q(X) be the space of quasicontinuous real-valued functions and τ_p be the topology of the pointwise convergence. For (Q(X), τ_p) complete metrizability, Polishness and Cech-completeness are equivalent. If (Q(X), τ_p) is completely metrizable, then X is countable and the set I(X) of isolated points of X is dense in X. If X is first countable, then (Q(X), τ_p) is completely metrizable if and only if X is countable and I(X) is dense in X.

math.GN↗

arXiv2026

Borel 1 type mappings and the respective equi-families

We investigate classes of functions from a topological space to a metric space that are related to those of Borel class 1. Following the idea defining an equi-Baire 1 family (due to Lecomte) we define the respective equi-families of functions from the considered classes. We observe that studying of equi-families can be reduced to the exploration of a single orbit map with values in a product space. We consider the closure of equi-families with respect to the topology of pointwise convergence. Finally, we investigate functions $f\colon X\times Y\to Z$, for metric spaces $X,Y,Z$, with sections that are equi-continuous, equi-Baire~1 or have equi-generalized Lebesgue property with respect to measurable sets of class $α$. In particular, we generalize a result of Grande.

math.GN↗

arXiv2018

Compactness in function spaces

Let $X$ be a locally compact topological space, $(Y,d)$ be a boundedly compact metric space and $LB(X,Y)$ be the space of all locally bounded functions from $X$ to $Y$. We characterize compact sets in $LB(X,Y)$ equipped with the topology of uniform convergence on compacta. From our results we obtain the following interesting facts for $X$ compact: $\bullet$ If $(f_n)_n$ is a sequence of uniformly bounded finitely equicontinuous functions of Baire class $α$ from $X$ to $\Bbb R$, then there is a uniformly convergent subsequence $(f_{n_k})_k$; $\bullet$ If $(f_n)_n$ is a sequence of uniformly bounded finitely equicontinuous lower (upper) semicontinuous functions from $X$ to $\Bbb R$, then there is a uniformly convergent subsequence $(f_{n_k})_k$; $\bullet$ If $(f_n)_n$ is a sequence of uniformly bounded finitely equicontinuous quasicontinuous functions from $X$ to $Y$, then there is a uniformly convergent subsequence $(f_{n_k})_k$.

math.GN↗

arXiv2018

Topological Properties of the Space of Convex Minimal Usco Maps

Let X be a Tychonoff space and MC(X) be the space of convex minimal usco maps with values in R, the space of real numbers. Such set-valued maps are important in the study of subdifferentials of convex functions. Using the strong Choquet game we prove complete metrizability of MC(X) with the upper Vietoris topology. If X is normal, elements of MC(X) can be approximated in the Vietoris topology by continuous functions. We also study first countability, second countability and other properties of the upper Vietoris topology on MC(X).

math.GN↗