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Homelocally homeomorphic
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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 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.

Nontrivial examples arise with locally Euclidean spaces, especially manifolds.
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