$X$ with a line(intersecting on one point) may not be homotopic equivalent to $X$?

Question

I felt weird when I tried to prove the statement of title.

$X$ with a line(intersecting on one point) may be homotopic equivalent to $X$

If $X$ is Hausdorff, it's possible to prove. I have a positive result of this.

However, for arbitrary topological space, how to prove it? Is there any counterexample of the statement?

Answer 1

The line with two origins does not deformation retract on two origins. Now take $X=\{a,b\}$ with indiscrete topology and add it a line. Then it is isomorphic to the line with two origins where two origins are $a$ and $b$.