What is an example of a topological space $(X,\tau)$ on more than one point, with the following properties?

1. the only homeomorphism $\varphi:X\to X$ is the identity, and
2. given $x,y\in X$ there are open sets $U, V$ with $x\in U, y\in V$ and $U\cong V$ with respect to the subspace topologies.