A+ CATEGORY SCIENTIFIC UNIT

PDF files of articles are only available for institutions which have paid for the online version upon signing an Institutional User License.

Construction of functions with given cluster sets

Volume 152 / 2018

Oleksandr V. Maslyuchenko, Denys P. Onypa Colloquium Mathematicum 152 (2018), 55-64 MSC: Primary 54C50; Secondary 54C60. DOI: 10.4064/cm6781-2-2017 Published online: 26 January 2018

Abstract

We continue our study of functions on the boundary of their domain and obtain some results on cluster sets of functions between topological spaces. For a compact space $\overline {Y}$ the cluster set of a function $f:D\to \overline {Y}$ at a point $x\in \overline {D}$ is the set $\overline {f}(x)=\bigcap \{\overline {f(U\cap D)}: U$ is a neighborhood of $x\}$ and it equals the $x$-section of the closure of the graph of $f$. We prove that for a metrizable topological space $X$, a dense subspace $Y$ of a metrizable compact space $\overline {Y}$, a closed nowhere dense subset $L$ of $X$, an upper continuous compact-valued multifunction ${\varPhi :L\multimap \overline {Y}}$ and a set $D\subseteq X\setminus L$ such that $L\subseteq \overline {D}$, there exists a function $f:D\to Y$ such that the cluster set $\overline {f} (x)$ is equal to $\varPhi (x)$ for any $x\in L$.

Authors

  • Oleksandr V. MaslyuchenkoInstitute of Mathematics
    Academia Pomeraniensis in Słupsk
    76-200 Słupsk, Poland
    e-mail
  • Denys P. OnypaDepartment of Mathematical Analysis
    Chernivtsi National University
    58012 Chernivtsi, Ukraine
    e-mail

Search for IMPAN publications

Query phrase too short. Type at least 4 characters.

Rewrite code from the image

Reload image

Reload image