206: 2 Points Are Topologically Path-Connected iff There Is Path That Connects 2 Points

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A description/proof of that 2 points are topologically path-connected iff there is path that connects 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: Description
• 2: Proof
• 3: None

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Starting Context

• The reader knows a definition of topological path-connected-ness of 2 points.
• The reader admits the proposition that any restriction of any continuous map on the domain and the codomain is continuous.
• The reader admits the proposition that any expansion of any continuous map on the codomain is continuous.

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Target Context

• The reader will have a description and a proof of 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.

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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, $T$, any points, $p_1,p_2\in T$, are path-connected if and only if there is a path, $\lambda :[r_1,r_2]\to T$, such that $\lambda (r_1)=p_1$ and $\lambda (r_2)=p_2$

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2: Proof

Suppose that there is a $\lambda$. Let us prove that $T_1:=\lambda ([r_1,r_2])$ is a path-connected topological subspace. For any points, $p_3,p_4\in T_1$, $p_3=\lambda (r_3)$ and $p_4=\lambda (r_4)$, and ${\lambda }^{\prime }:[r_3,r_4]\to T_1$, which is the restriction of $\lambda$ on the domain and the codomain, is a path on $T_1$, by the proposition that any restriction of any continuous map on the domain and the codomain is continuous. So, $p_3$ and $p_4$ are path-connected on $T_1$. So, $p_1$ and $p_2$ are path-connected on $T$, as there is a path-connected topological subspace that contains the both points.

Suppose that $p_1$ and $p_2$ are path-connected. There is a path-connected topological subspace, $p_1,p_2\in T_1\subseteq T$. There is a path, ${\lambda }^{\prime }:[r_1,r_2]\to T_1$, that connects the 2 points, but by the proposition that any expansion of any continuous map on the codomain is continuous, $\lambda :[r_1,r_2]\to T$ is a path on $T$.

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3: None

This proposition may be a definition for some people, but as I have adopted another definition, this proposition is due.

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References

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