199: Topological Path-Connected-Ness of 2 Points

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A definition of topological path-connected-ness of 2 points

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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: Definition
• 2: Note

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Starting Context

• The reader knows a definition of topological path.

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Target Context

• The reader will have a definition of path-connected-ness of any 2 points on any topological space.

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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: Definition

For any topological space, $T$, any points, $p_1,p_2\in T$, are path-connected if there is a path-connected topological subspace, $T_1\subseteq T$, that contains the 2 points

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2: Note

Although there can be another, probably more plain, definition that requires just a path on $T$ that connects the 2 points, it is equivalent to the definition of this article, by the proposition that any 2 points are path-connected on any topological space if and only if there is a path that connects the 2 points on the topological space.

The definition of this article is adopted in order to be more symmetric with the definition of topological connected-ness of 2 points.

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References

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