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I have difficulty coming up with a proof that if two points in a topological space $X$ are connected by a path, then there is an injective continuous map of $[0,1]$ to $X$ that sends $0$ and $1$ to those two points. Is this true? How is this proved?

Edit: As answered by Zev Chonoles, this is obviously false for general spaces. I would also like to have an answer for Hausdorff spaces.

Update: it turns out the hard part of this question is a duplicate of A question about path-connected and arcwise-connected spaces

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Alexey
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Let $X=\{a,b\}$ with the topology $T=\{\varnothing,\{a\},X\}$ (which is non-Hausdorff). The map $f:[0,1]\to X$ defined by $$f(t)=\begin{cases} a & \text{ if }t\in [0,1),\\ b & \text{ if }t=1 \end{cases}$$ is continuous, but there is clearly no injective map from $[0,1]$ to $X$.

Your conjecture is true if we require $X$ to be Hausdorff. According to Wikipedia, a path-connected Hausdorff space is necessarily arc-connected. Thus, if $f:[0,1]\to X$ is the path from $a$ to $b$, the subspace $f([0,1])$ of $X$ is path-connected and Hausdorff, hence arc-connected, hence there is an arc connecting $a$ and $b$, which is in particular an injective continuous path $[0,1]\to X$.

Zev Chonoles
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  • Thanks, it was obvious! Excuse me for not accepting your answer because i am more interested in Hausdorff spaces, i am going to edit the question. – Alexey Mar 06 '13 at 08:27
  • Thanks for pointing me to arc connected spaces, i forgot that those are different terms. I have also found that this question is a duplicate. If i find somewhere a link to a proof, i will accept your answer, otherwise i am interested in a link to a proof. – Alexey Mar 06 '13 at 09:03