$[0,1]$ is topologically same as $[-1,1]$ mean?

Question

I am just getting started with topology. What does $[0,1]$ topologically same as $[-1,1]$ mean?

Any geometric intuition would be great!

Answer 1

You can just stretch the interval $[0,1]$ to make it into $[-1,1]$ in a continuous way, formally you can do it with a linear map, like $x \to 2x-1$, which has a continuous inverse too. In fact, all closed intervals $[a,b]$ are the same, in the sense that one can be transformed to another and back. Topologically it's the only connected, locally connected compact Hausdorff space $X$ with a countable base such that $X\setminus \{x\}$ is disconnected, except for two points $x$ (the endpoints).