200: Topological Path-Connected-ness of 2 Points Is Equivalence Relation

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A description/proof of that topological path-connected-ness of 2 points is equivalence relation

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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 topological path-connected-ness of 2 points.
• The reader knows a definition of equivalence relation.
• The reader admits 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 reader admits 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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Target Context

• The reader will have a description and a proof of the proposition that topological path-connected-ness of 2 points is an equivalence relation.

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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, $T$, path-connected-ness of 2 points is an equivalence relation.

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2: Proof

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, we only need to cite a path on $T$ in order to prove that 2 concerned points are path-connected.

For any point, $p\in T$, $p$ and $p$ are path-connected, because the curve, $\lambda :[0,1]\to T$, which constantly maps to $p$ is a path.

For any points, $p_1,p_2\in T$, that are path-connected, $p_2$ and $p_1$ are path-connected, because there is a path, $\lambda :[0,1]\to T$, where $\lambda (0)=p_1$ and $\lambda (1)=p_2$, so, there is the path, ${\lambda }^{\prime }:[0,1]\to [0,1]\to T$, where its 1st-half, $[0,1]\to [0,1]$, is $r\mapsto 1-r$ and its 2nd-half, $[0,1]\to T$, is $\lambda$, continuous as a compound of continuous maps, with ${\lambda }^{\prime }(0)=p_2$ and ${\lambda }^{\prime }(1)=p_1$.

For any points, $p_1,p_2,p_3\in T$, such that $p_1$ and $p_2$ are path-connected and $p_2$ and $p_3$ are path-connected, there are paths, ${\lambda }_1:[0,2^{-1}]\to T$ where ${\lambda }_1(0)=p_1$ and ${\lambda }_1(2^{-1})=p_2$ and ${\lambda }_2:[2^{-1},1]\to T$ where ${\lambda }_2(2^{-1})=p_2$ and ${\lambda }_2(1)=p_3$. Let us define ${\lambda }^{\prime }:[0,1]\to T$ as ${\lambda }^{\prime }{|}_{[0,2^{-1}]}={\lambda }_1$ and ${\lambda }^{\prime }{|}_{[2^{-1},1]}={\lambda }_2$. ${\lambda }^{\prime }$ is continuous by 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, and ${\lambda }^{\prime }(0)=p_1$ and ${\lambda }^{\prime }(1)=p_3$. So, $p_1$ and $p_3$ are path-connected.

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References

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