207: 2 Points That Are Path-Connected on Topological Subspace Are Path-Connected on Larger Subspace

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A description/proof of that 2 points that are path-connected on topological subspace are path-connected on larger subspace

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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 path-connected-ness of 2 points.
• 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 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 that are path-connected on any topological subspace are path-connected on any larger subspace.

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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 subspace, $T_1\subseteq T$, and any 2 points, $p_1,p_2\in T_1$, that are path-connected on $T_1$, $p_1$ and $p_2$ are path-connected on any larger subspace, $T_2$, such that $T_1\subseteq T_2\subseteq T$.

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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, there is a path, $\lambda :[0,1]\to T_1$ such that $\lambda (0)=p_1$ and $\lambda (1)=p_2$. $\lambda ([0,1])\subseteq T_1\subseteq T_2$. ${\lambda }^{\prime }:[0,1]\to T_2$ as the expansion of $\lambda$ on the codimension is a path on $T_2$, by the proposition that any expansion of any continuous map on the codomain is continuous. 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, $p_1$ and $p_2$ are path-connected on $T_2$.

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References

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