<abstract xmlns="http://www.w3.org/1999/xhtml">

<p>The paper tries to survey the recent results about relationships between covering properties of a topological space <italic>X</italic> and the space USC(<italic>X</italic>) of upper semicontinuous functions on <italic>X</italic> with the topology of pointwise convergence. Dealing with properties of continuous functions C(<italic>X</italic>), we need shrinkable covers. The results are extended for <italic>A</italic>-measurable and upper <italic>A</italic>-semimeasurable functions where <italic>A</italic> is a family of subsets of <italic>X</italic>. Similar results for covers respecting a bornology and spaces USC(<italic>X</italic>) or C(<italic>X</italic>) endowed by a topology defined by using the bornology are presented. Some of them seem to be new.</p>
</abstract>

The paper tries to survey the recent results about relationships between covering properties of a topological space X and the space USC(X) of upper semicontinuous functions on X with the topology of pointwise convergence. Dealing with properties of continuous functions C(X), we need shrinkable covers. The results are extended for A-measurable and upper A-semimeasurable functions where A is a family of subsets of X. Similar results for covers respecting a bornology and spaces USC(X) or C(X) endowed by a topology defined by using the bornology are presented. Some of them seem to be new.

Real Functions, Covers and Bornologies

Tatra Mountains Mathematical Publications

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

{"article-title":"Real Functions, Covers and Bornologies"}

The paper tries to survey the recent results about relationships between covering properties of a topological space X and the space USC(X) of upper...

Real Functions, Covers and Bornologies

Published Online: Jan 01, 2022

Page range: 199 - 214

Received: Feb 01, 2021

DOI: https://doi.org/10.2478/tmmp-2021-0014

Keywords
cover, upper semicontinuous function, measurable function, upper semimeasurable function, bornology

© 2021 Mathematical Institute, Slovak Academy of Sciences

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

Real Functions, Covers and Bornologies

Published Online: Jan 01, 2022

Page range: 199 - 214

Received: Feb 01, 2021

DOI: https://doi.org/10.2478/tmmp-2021-0014

Keywordscover, upper semicontinuous function, measurable function, upper semimeasurable function, bornology

© 2021 Mathematical Institute, Slovak Academy of Sciences

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

Keywords
cover, upper semicontinuous function, measurable function, upper semimeasurable function, bornology