If $(X,T)$ is a topological space, are all element of $T$ homeomorphic to $X$?

Question

I know that all open set $U$ of $\mathbb R^n$ is homeomorphic to $\mathbb R^n$. Is this property true in all topological space ? I mean, if $(X,T)$ is a topological space, then all element $U\in T$ are homeomorphic to $X$?

Answer 1

You can't know that, since it is not true. For instance, $(-1,1)\setminus\{0\}$ is not homeomorphic to $\mathbb R$.

Answer 2

No. Take the simplest possible case: a set with two members, {a, b} and the "discrete topology", {{},{a},{b}, {a,b}}. Two of the sets in the topology contain only one member while the entire space contains two members.

In fact, your original statement "all open set U in $R^n$ is homeomorphic to $R^n$" is NOT true! One of the opens sets is the empty set and the empty set is definitely not homeomorphic to $R^n$.