318: Topological Sum of Paracompact Topological Spaces Is Paracompact

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A description/proof of that topological sum of paracompact topological spaces is paracompact

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Topics

About: topological space

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The table of contents of this article

• Starting Context
• Target Context
• Orientation
• Main Body
• 1: Description
• 2: Proof
• 3: Note

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Starting Context

• The reader knows a definition of topological sum.
• The reader knows a definition of paracompact topological space.
• The reader admits the proposition that the topological sum of any possibly uncountable number of Hausdorff topological spaces is Hausdorff.

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Target Context

• The reader will have a description and a proof of the proposition that the topological sum of any possibly uncountable number of paracompact topological spaces is paracompact.

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Orientation

There is a list of definitions discussed so far in this site.

There is a list of propositions discussed so far in this site.

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Main Body

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1: Description

For any paracompact topological spaces, $\{T_{\alpha }|\alpha \in A\}$ where $A$ is a possibly uncountable indices set, the topological sum, $T:=\coprod _{\alpha }{T}_{\alpha }$, is paracompact.

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2: Proof

$T$ is a Hausdorff topological space as the topological sum of Hausdorff topological spaces, by the proposition that the topological sum of any possibly uncountable number of Hausdorff topological spaces is Hausdorff.

For any open cover, $S:=\{U_{\beta }\subseteq T|\beta \in B\}$ where $B$ is any possibly uncountable indices set, of $T$, $S_{\alpha }:=\{U_{\beta }\cap T_{\alpha }|\beta \in B\}$ is an open cover of $T_{\alpha }$. As $T_{\alpha }$ is paracompact, there is a locally finite refinement, ${S_{\alpha }}^{\prime }:=\{{U_{\alpha -\gamma }}^{\prime }\subseteq U_{\beta }\cap T_{\alpha }|\gamma \in C\}$ where $C$ is a possibly uncountable indices set. ${\cup }_{\alpha }{S_{\alpha }}^{\prime }$ is a locally finite refinement of $S$, because ${U_{\alpha -\gamma }}^{\prime }$ is open on $T$, ${\cup }_{\alpha }{S_{\alpha }}^{\prime }$ covers $T$, ${U_{\alpha -\gamma }}^{\prime }\subseteq U_{\beta }$, and around any point, $p\in T$, $p\in T_{\alpha }$, and there is a neighborhood, $N_p$, contained in $T_{\alpha }$ that intersects only finite elements of ${S_{\alpha }}^{\prime }$, because $T_{\alpha }$ is open and disjoint from any $T_{\beta }$ such that $\beta \ne \alpha$.

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3: Note

When $T$ is any Hausdorff topological space that is the disjoint union of any possibly uncountable number of open paracompact subspaces, $T$ is paracompact, because $T$ is the topological sum of the subspaces.

The subspaces have to be open and disjoint in order to apply this proposition, because otherwise, $T$ would not be any topological sum.

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References

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