Why $[0,3]/(1,2)$ is not homeomorphism to $[0,1]$

Question

Clearly $[0,3]/(1,2)=[0,1]+ {\rm point} A+[2,3]$. How can I prove those are not homeomorphic

Answer 1

The set $\{A\}$, where $A$ is the identified set $(1,2)$, seen as an equivalence class, so a point in the quotient $Y:=[0,3]{/}(1,2)$ (under the identification map $q:[0,3]\to Y$), is not closed as $q^{-1}[\{A\}]=(1,2)$ is not closed in $[0,3]$. So $Y$ is not $T_1$ and $[0,3]$ is, being metric.

So they're not homeomorphic.

Note that $Y':=[0,3]{/}[1,2]$ is homeomorphic to $[0,1]$.