Let $X$ and $Y$ be topological spaces. Then $X$ is locally homeomorphic to $Y$ if for every $x\in X$ there is a neighbourhood $U\subseteq X$ of $x$ and an open set $V\subseteq Y$ such that $U$ and $V$ with their respective subspace topology are homeomorphic.

Examples

• Let $X=\{1\}$ and $Y=\{2,3\}$ be discrete spaces with one resp. two elements. Since $X$ and $Y$ have different cardinalities, they cannot be homeomorphic. They are, however, locally homeomorphic to each other.
• Again, let $X=\{1\}$ be a discrete space with one element, but now let $Y=\{2,3\}$ the space with topology $\{\emptyset,\{2\},Y\}$ Then $X$ is still locally homeomorphic to $Y$ but $Y$ is not locally homeomorphic to $X$ since the smallest neighbourhood of $3$ already has more elements than $X$
• Now, let $X$ be as in the previous examples, and $Y=\{2,3\}$ be indiscrete. Then neither $X$ is locally homeomorphic to $Y$ nor the other way round.
• Non-trivial examples arise with locally Euclidean spaces, especially manifolds.