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Suppose $(X,T_X)$ is a topological space and $\infty_X \notin X$. Write $X^* = X \cup \{\infty_X\}$ and suppose the open sets of $X^*$ the empty set and the union of an open set in $X$ and the point $\{\infty_X\}$.

I want to show that $(X^*,T_X^*)$ is a path connected space. This means I have to show that there is a single path connected component ($X^*$ itself), i.e. that for any $x,y \in X^*$ there is a continuous map $\gamma: I \rightarrow X$ where $\gamma(0) = x, \gamma(1)=y$. Here is $I=[0,1]$ equipped with the Euclidean topology.

I don't really have a clue how to proceed, should I do something with the fact that $\gamma$ is continuous and work with the definition? I'm guessing that the special point $\{\infty_X\}$ has an important role to show path connectedness but I am confused because we don't have any information on the connectedness of $X$.

EDIT: Can we conclude from this that any topological space can be embedded as a closed subspace of a path connected topological space? If so, how would I go about doing that?

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Take $x\in X$ and define $\gamma\colon[0,1]\longrightarrow X$ by$$\gamma(t)=\begin{cases}\infty_X&\text{ if }t\in[0,1)\\x&\text{ otherwise.}\end{cases}$$Then $\gamma(0)=\infty_X$ and $\gamma(1)=x$. Besides, $\gamma$ is continuus since, if $A$ is an open subset of $X$:

  • If $\infty_X,x\notin A$, $\gamma^{-1}(A)=\emptyset$, which is an open subset of $[0,1]$;
  • If $\infty_X\in A$ and $x\notin A$, $\gamma^{-1}(A)=[0,1)$, which is an open subset of $[0,1]$;
  • If $\infty_X,x\in A$, $\gamma^{-1}(A)=[0,1]$, which is an open subset of $[0,1]$.

Since you cannot have $x\in A$ and $\infty_X\notin A$, this proves that $\gamma$ is continuous. And $\gamma$ is a path from $\infty_X$ to $x$. Since there is such a path for every $x\in X$, $X$ is path-connected.

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  • $\begingroup$ What about $x,y \neq \infty_X$? It seems to me you only constructed paths with $\infty_X$... $\endgroup$ Mar 26 at 20:50
  • $\begingroup$ You take a path from $x$ to $\infty_X$, followed by a path from $\infty_X$ to $y$. $\endgroup$ Mar 26 at 21:22
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    $\begingroup$ I'm glad I could help. $\endgroup$ Mar 26 at 21:35
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    $\begingroup$ Or directly, $\gamma(0)=x$, $\gamma(1)=y$, and $\gamma(t)=\infty_X$ otherwise $\endgroup$ Mar 26 at 21:48
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    $\begingroup$ The continuous injection is the identity map from $X$ into $X^*$ ($x\mapsto x$). $\endgroup$ Mar 28 at 19:27

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