locally homeomorphic
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 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 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^{} $\{\mathrm{\varnothing},\{2\},Y\}$. 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 http://planetmath.org/node/3120indiscrete. Then neither $X$ is locally homeomorphic to $Y$ nor the other way round.

•
Nontrivial examples arise with locally Euclidean spaces, especially manifolds.
Title  locally homeomorphic 

Canonical name  LocallyHomeomorphic 
Date of creation  20130322 15:14:34 
Last modified on  20130322 15:14:34 
Owner  GrafZahl (9234) 
Last modified by  GrafZahl (9234) 
Numerical id  4 
Author  GrafZahl (9234) 
Entry type  Definition 
Classification  msc 5400 
Synonym  local homeomorphy 
Related topic  LocallyEuclidean 