388: Map Between Topological Spaces Is Continuous if Domain Restriction of Map to Each Closed Set of Finite Closed Cover is Continuous

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A description/proof of that map between topological spaces is continuous if domain restriction of map to each closed set of finite closed cover is continuous

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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

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Starting Context

• The reader knows a definition of continuous map.
• The reader knows a definition of closed set.
• The reader admits the proposition that any closed set on any closed topological subspace is closed on the base space.
• The reader admits the proposition that if the preimage of any closed set under a topological spaces map is closed, the map is continuous.

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Target Context

• The reader will have a description and a proof of the proposition that any map between topological spaces is continuous if the domain restriction of the map to each closed set of a finite closed cover is continuous.

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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 topological spaces, $T_1,T_2$, and any map, $f:T_1\to T_2$, if there is a finite closed cover of $T_1$, $\{C_i\subseteq T_1\},{\cup }_i{C}_i=T_1$, such that each $f{|}_{C_i}:C_i\to T_2$ is continuous, $f$ is continuous.

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2: Proof

For any closed set, $C\subseteq T_2$, ${f{|}_{C_i}}^{-1}(C)$ is closed on $C_i$, and on $T_1$, by the proposition that any closed set on any closed topological subspace is closed on the base space. $f^{-1}(C)={\cup }_i{f{|}_{C_i}}^{-1}(C)$, because for any $p\in f^{-1}(C)$, $f(p)\in C$, but $p\in {\cup }_i{C}_i$, so, $f{|}_{C_i}(p)\in C$ for an $i$, $p\in {f{|}_{C_i}}^{-1}(C)$; for any $p\in {\cup }_i{f{|}_{C_i}}^{-1}(C)$, $p\in {f{|}_{C_i}}^{-1}(C)$ for an $i$, so, $f{|}_{C_i}(p)\in C$, so, $f(p)\in C$, so, $p\in f^{-1}(C)$. So, $f^{-1}(C)$ is close as the finite union of closed sets. By the proposition that if the preimage of any closed set under a topological spaces map is closed, the map is continuous, $f$ is continuous.

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References

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