Let us call a topological space $(X,\tau)$ $\mathbb{R}$-*like* if it is homogeneous, connected, $T\_2$, has a basis consisting of open sets homeomorphic to $X$, and $|X|>1$.


What is an example of an $\mathbb{R}$-like space that is not homeomorphic to some power of $\mathbb{R}$?



$\mathbb{R}$-like spaces
An example is $\mathbb{R}^2\setminus\{0\}$


