Journal of Formalized Mathematics
Volume 4, 1992
University of Bialystok
Copyright (c) 1992 Association of Mizar Users

## Continuity of Mappings over the Union of Subspaces

Zbigniew Karno
Warsaw University, Bialystok

### Summary.

Let $X$ and $Y$ be topological spaces and let $X_{1}$ and $X_{2}$ be subspaces of $X$. Let $f : X_{1} \cup X_{2} \rightarrow Y$ be a mapping defined on the union of $X_{1}$ and $X_{2}$ such that the restriction mappings $f_{\mid X_{1}}$ and $f_{\mid X_{2}}$ are continuous. It is well known that if $X_{1}$ and $X_{2}$ are both open (closed) subspaces of $X$, then $f$ is continuous (see e.g. [7, p.106]). \par The aim is to show, using Mizar System, the following theorem (see Section 5): {\em If $X_{1}$ and $X_{2}$ are weakly separated, then $f$ is continuous} (compare also [14, p.358] for related results). This theorem generalizes the preceding one because if $X_{1}$ and $X_{2}$ are both open (closed), then these subspaces are weakly separated (see ). However, the following problem remains open. \begin{itemize} \item[ ] {\bf Problem 1.} Characterize the class of pairs of subspaces $X_{1}$ and $X_{2}$ of a topological space $X$ such that ($\ast$) for any topological space $Y$ and for any mapping $f : X_{1} \cup X_{2} \rightarrow Y$, $f$ is continuous if the restrictions $f_{\mid X_{1}}$ and $f_{\mid X_{2}}$ are continuous. \end{itemize} In some special case we have the following characterization: {\em $X_{1}$ and $X_{2}$ are separated iff $X_{1}$ misses $X_{2}$ and the condition} ($\ast$) {\em is fulfilled.} In connection with this fact we hope that the following specification of the preceding problem has an affirmative answer. \begin{itemize} \item[ ] {\bf Problem 2.} Suppose the condition ($\ast$) is fulfilled. Must $X_{1}$ and $X_{2}$ be weakly separated ? \end{itemize} Note that in the last section the concept of the union of two mappings is introduced and studied. In particular, all results presented above are reformulated using this notion. In the remaining sections we introduce concepts needed for the formulation and the proof of theorems on properties of continuous mappings, restriction mappings and modifications of the topology.

#### MML Identifier: TMAP_1

The terminology and notation used in this paper have been introduced in the following articles             

#### Contents (PDF format)

1. Set-Theoretic Preliminaries
2. Selected Properties of Subspaces of Topological Spaces
3. Continuity of Mappings
4. A Modification of the Topology of Topological Spaces
5. Continuity of Mappings over the Union of Subspaces
6. The Union of Continuous Mappings

#### Acknowledgments

I would like to express my thanks to Professors A.~Trybulec and Cz.~Byli\'nski for their active interest in the publication of this article and for elucidating to me new advanced Mizar constructions.

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