330: Inclusion into Topological Space from Subspace Is Continuous

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A description/proof of that inclusion into topological space from subspace 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 subspace topology.
• The reader knows a definition of continuous map.

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Target Context

• The reader will have a description and a proof of the proposition that for any topological space, the inclusion from any subspace into the topological space 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 space, $T_1$, and any subspace, $T_2\subseteq T_1$, the inclusion, $f:T_2\to T_1$ is continuous.

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2: Proof

For any open set, $U\in T_1$, $f^{-1}(U)=U\cap T_2$, which is open on $T_2$, by the definition of subspace topology.

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References

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