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

## On a Duality Between Weakly Separated Subspaces of Topological Spaces

Zbigniew Karno
Warsaw University, Bialystok

### Summary.

Let $X$ be a topological space and let $X_{1}$ and $X_{2}$ be subspaces of $X$ with the carriers $A_{1}$ and $A_{2}$, respectively. Recall that $X_{1}$ and $X_{2}$ are {\em weakly separated}\/ if $A_{1} \setminus A_{2}$ and $A_{2} \setminus A_{1}$ are separated (see  and also  for applications). Our purpose is to list a number of properties of such subspaces, supplementary to those given in . Note that in the Mizar formalism the carrier of any topological space (hence the carrier of any its subspace) is always non-empty, therefore for convenience we list beforehand analogous properties of weakly separated subsets without any additional conditions.\par To present the main results we first formulate a useful definition. We say that $X_{1}$ and $X_{2}$ {\em constitute a decomposition}\/ of $X$ if $A_{1}$ and $A_{2}$ are disjoint and the union of $A_{1}$ and $A_{2}$ covers the carrier of $X$ (comp. ). We are ready now to present the following duality property between pairs of weakly separated subspaces~: {\em If each pair of subspaces $X_{1}$, $Y_{1}$ and $X_{2}$, $Y_{2}$ of $X$ constitutes a decomposition of $X$, then $X_{1}$ and $X_{2}$ are weakly separated iff $Y_{1}$ and $Y_{2}$ are weakly separated}. From this theorem we get immediately that under the same hypothesis, {\em $X_{1}$ and $X_{2}$ are separated iff $X_{1}$ misses $X_{2}$ and $Y_{1}$ and $Y_{2}$ are weakly separated}. Moreover, we show the following enlargement theorem~: {\em If $X_{i}$ and $Y_{i}$ are subspaces of $X$ such that $Y_{i}$ is a subspace of $X_{i}$ and $Y_{1} \cup Y_{2} = X_{1} \cup X_{2}$ and if $Y_{1}$ and $Y_{2}$ are weakly separated, then $X_{1}$ and $X_{2}$ are weakly separated}. We show also the following dual extenuation theorem~: {\em If $X_{i}$ and $Y_{i}$ are subspaces of $X$ such that $Y_{i}$ is a subspace of $X_{i}$ and $Y_{1} \cap Y_{2} = X_{1} \cap X_{2}$ and if $X_{1}$ and $X_{2}$ are weakly separated, then $Y_{1}$ and $Y_{2}$ are weakly separated}. At the end we give a few properties of weakly separated subspaces in subspaces.

#### MML Identifier: TSEP_2

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

#### Contents (PDF format)

1. Certain Set--Decompositions of a Topological Space
2. Duality between Pairs of Weakly Separated Subsets
3. Certain Subspace--Decompositions of a Topological Space
4. Duality between Pairs of Weakly Separated Subspaces

#### Bibliography

 Zbigniew Karno. Continuity of mappings over the union of subspaces. Journal of Formalized Mathematics, 4, 1992.
 Zbigniew Karno. Separated and weakly separated subspaces of topological spaces. Journal of Formalized Mathematics, 4, 1992.
 Kazimierz Kuratowski. \em Topology, volume I. PWN - Polish Scientific Publishers, Academic Press, Warsaw, New York and London, 1966.
 Beata Padlewska and Agata Darmochwal. Topological spaces and continuous functions. Journal of Formalized Mathematics, 1, 1989.
 Andrzej Trybulec. Tarski Grothendieck set theory. Journal of Formalized Mathematics, Axiomatics, 1989.
 Andrzej Trybulec. A Borsuk theorem on homotopy types. Journal of Formalized Mathematics, 3, 1991.
 Zinaida Trybulec. Properties of subsets. Journal of Formalized Mathematics, 1, 1989.
 Miroslaw Wysocki and Agata Darmochwal. Subsets of topological spaces. Journal of Formalized Mathematics, 1, 1989.

Received November 9, 1992

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