Let $X_1$ and $X_2$ be subspaces of $\mathbb{R}$ given by $X_1 = (0,1)\cup(3,4)$ and $X_2 = (0,1)\cup(1,2)$. Is $X_1 \cong X_2 $?

Question

Correct me if im wrong, but intuitively i think that it's not a homeomorphism between $X_1 \cong X_2 $, nevertheless i need a more rigorous answer to this. Anyone who has a suggestion or idea please share it.

Answer 1

Put $$f(x)=\left\{\begin{matrix} x & \text{if}~~~ x\in (0,1),\\ x+2 &\text{if}~~~ x\in (1,2) \end{matrix}\right.$$

is a continuous bijection from $(0,1)\cup (1,2)$ to $(0,1)\cup (3,4),$ with a continuous inverse

$$f^{-1}(x)=\left\{\begin{matrix} x & \text{if}~~~ x\in (0,1),\\ x-2 &\text{if}~~~ x\in (3,4). \end{matrix}\right.$$

And so $(0,1)\cup (1,2)$ and $(0,1)\cup (3,4)$ are homeomorphic.