# If two points in a space are connected by a path, is there an injective map of an interval?

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

• General spaces? Hausdorff spaces? $T_1$ spaces? – Asaf Karagila Mar 6 '13 at 8:20
• As stated, i am asking about general spaces. In fact, i am more interested in Hausdorff spaces. – Alexey Mar 6 '13 at 8:23

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 $[0,1]\to X$.