Locally Homeomorphism

A topological space $X$ is locally homeomorphic to $Y$ if every point of $X$ has a neighborhood that is homeomorphic to an open subset of $Y$. 

Fomral Definition

A function $f:X\to Y$ between two topological spaces is called a local homeomorphism if every point $x\in X$ has an open neighborhood $U$ whose image $f\left(U\right)$ is open in $Y$ and the restriction $f{|}_U:U\to f\left(U\right)$ is a homeomorphism(with subspace topology).

Example

For example, a Manifold of dimension $n$ is locally homeomorphic to $R^n$.

Property

If there is a local homeomorphism from $X$ to $Y$, then $X$ is locally homeomorphic to $Y$, but the converse is not always true. For example, $S^2$, being a manifold, is locally homeomorphic to the plane ${\mathbb{R}}^2$, but there is no local homeomorphism $S^2\to {\mathbb{R}}^2$.

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