393: Restriction of Continuous Map on Domain and Codomain Is Continuous

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A description/proof of that restriction of continuous map on domain and codomain 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 topological space.
• The reader knows a definition of continuous map.
• The reader knows a definition of subspace topology.

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Target Context

• The reader will have a description and a proof of the proposition that any restriction of any continuous map on the domain and the codomain 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, $T_1,T_2$, any continuous map, $f:T_1\to T_2$, any subset, $S_1\subseteq T_1$, and any subset, $S_2\subseteq T_2$ such that $f(S_1)\subseteq S_2$, $f{|}_{S_1}:S_1\to S_2$ is continuous.

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2: Proof

For any open set, $U\subseteq S_2$, $U=U^{\prime }\cap S_2$ where $U^{\prime }\subseteq T_2$ is open on $T_2$, by the definition of subspace topology. ${f{|}_{S_1}}^{-1}(U)={f{|}_{S_1}}^{-1}(U^{\prime }\cap S_2)=f^{-1}(U^{\prime })\cap S_1$, because for any $p\in {f{|}_{S_1}}^{-1}(U^{\prime }\cap S_2)$, $f{|}_{S_1}(p)\in U^{\prime }\cap S_2$ by the definition of preimage, $f(p)\in U^{\prime }\cap S_2\subseteq U^{\prime }$, so, $p\in f^{-1}(U^{\prime })$, while of course $p\in S_1$ because $S_1$ is the domain of ${f{|}_{S_1}}^{-1}$; for any $p\in f^{-1}(U^{\prime })\cap S_1$, $f{|}_{S_1}(p)\in U^{\prime }$ because as $p$ is on $S_1$, $f{|}_{S_1}$ can operate on it with the same result, which is on $U^{\prime }$, but as $f{|}_{S_1}(S_1)=f(S_1)\subseteq S_2$, $f{|}_{S_1}(p)\in U^{\prime }\cap S_2$, so, $p\in {f{|}_{S_1}}^{-1}(U^{\prime }\cap S_2)$. As $f$ is continuous, $f^{-1}(U^{\prime })$ is open on $T_1$, and $f^{-1}(U^{\prime })\cap S_1$ is open on $S_1$ by the definition of subspace topology.

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References

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