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