463: 2 Points on Different Connected Components Are Not Path-Connected

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A description/proof of that 2 points on different connected components are not path-connected

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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 connected topological component.
• The reader knows a definition of topological path-connected-ness of 2 points.
• The reader admits the proposition that 2 points are topologically path-connected iff there is path that connects 2 points.
• The reader admits the proposition that any continuous map image of any connected subspace of the domain is connected on the codomain.
• The reader admits the proposition that any connected subspace of any topological space is contained in a connected component.

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Target Context

• The reader will have a description and a proof of the proposition that for any topological space with multiple connected components, any 2 points on any different connected components are not path-connected.

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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, $T$, with multiple connected components, $\{S_{\alpha }\}$, any 2 points, $p_1\in S_{{\alpha }^{\prime }}$ and $p_2\in S_{{\alpha }^{″}}$ where ${\alpha }^{\prime }\ne {\alpha }^{″}$ are not path-connected on $T$.

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2: Proof

Let us suppose that $p_1$ and $p_2$ were path-connected on $T$. There would be a path, $\lambda :I\to T$, such that $\lambda (0)=p_1$ and $\lambda (1)=p_2$, by the proposition that 2 points are topologically path-connected iff there is path that connects 2 points. $I$ would be connected and $\lambda (I)$ would be connected on $T$, by the proposition that any continuous map image of any connected subspace of the domain is connected on the codomain. As $\lambda (I)$ would not be contained in any 1 connected component, $\lambda (I)$ would not be connected on $T$, by the proposition that any connected subspace of any topological space is contained in a connected component, a contradiction. So, $p_1$ and $p_2$ are not path-connected on $T$.

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References

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