locally homeomorphic

Let $X$ and $Y$ be topological spaces. Then $X$ is locallyhomeomorphic to $Y$, if for every $x\in X$ there is a neighbourhood$U\subseteq X$ of $x$ and an http://planetmath.org/node/380open 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 homeomorphicto each other.

• Again, let $X=\{1\}$ be a discrete space with one element, butnow let $Y=\{2,3\}$ the space with topology$\{\mathrm{\varnothing },\{2\},Y\}$. Then $X$ is still locally homeomorphic to$Y$, but $Y$ is not locally homeomorphic to $X$, since the smallestneighbourhood of $3$ already has more elements than $X$.

• Now, let $X$ be as in the previous examples, and $Y=\{2,3\}$be http://planetmath.org/node/3120indiscrete. Then neither $X$ is locally homeomorphic to $Y$ northe other way round.

• Non-trivial examples arise with locally Euclidean spaces,especially manifolds.