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

1. the only homeomorphism from $X$ to itself is the identity, and
2. given $x,y\in X$ there are open sets $U, V$ with $x\in U, y\in V$ and a homeomorphism $\varphi: U\to V$ such that $\varphi(x) = y$.

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

1. the only homeomorphism from $X$ to itself is the identity, and
2. given $x,y\in X$ there are open sets $U, V$ with $x\in U, y\in V$ and a homeomorphism $\varphi: U\to V$ such that $\varphi(x) = y$, where $U,V$ are endowed with their respective subspace topologies.

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.

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

1. the only homeomorphism from $X$ to itself is the identity, and
2. given $x,y\in X$ there are open sets $U, V$ with $x\in U, y\in V$ and a homeomorphism $\varphi: U\to V$ such that $\varphi(x) = y$.