Locally homeomorphic spaces

Let $X$ and $Y$ be topological spaces. we say that $X$ is locally homeomorphic to $Y$, if for every $x\in X$ there is a neighborhood $U\subset X$ of $x$ and an open set $V\subset Y$, such that $U$ and $V$ are homeomorphic with respect to their relative topology.

Examples.

• Finite discrete spaces are always locally homeomorphic but not homeomorphic unless they have the same number of elements.
• Manifolds of the same dimension are locally homeomorphic.