212: Connected Component Is Open on Locally Connected Topological Space

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A description/proof of that connected component is open on locally connected topological space

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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 locally connected topological space.
• The reader admits the local criterion for openness.

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Target Context

• The reader will have a description and a proof of the proposition that for any locally connected topological space, each connected component is open.

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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 locally connected topological space, $T$, each connected component, $S$, is open.

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2: Proof

At any point, $p\in S$, take any neighborhood of $p$, $U_p$, on $T$. There is a connected neighborhood of $p$, ${U_p}^{\prime }$, contained in $U_p$. As ${U_p}^{\prime }$ is a connected subspace that contains $p$, ${U_p}^{\prime }\subseteq S$. By the local criterion for openness, $S$ is open.

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References

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