Definition
A local homeomorphism is a continuous map $p : E \rightarrow B$ between topological spaces such that for every point $b \in B$ there exists an open neighbourhood $U$ of $b$ with the property that $p^{-1}(U)$ is a (possibly empty) disjoint union of open sets in $E$, each of which is mapped homeomorphically onto $U$ by the restriction of $p$. Formally, for each $b \in B$ there is an open set $U \subseteq B$ with $b \in U$ and a family ${V_i}_{i \in I}$ of open subsets of $E$ such that
$$ p^{-1}(U)=\bigsqcup_{i\in I} V_i,\qquad p|_{V_i}: V_i \longrightarrow U \text{ is a homeomorphism for every } i. $$
Overview
Local homeomorphisms generalize the notion of covering maps by relaxing the global “evenly covering” condition: a covering map requires the family ${V_i}$ to be all the connected components of $p^{-1}(U)$ and to be uniform in cardinality across the base space. In contrast, a local homeomorphism only demands the existence of such neighbourhoods locally around each point, without any restriction on how many sheets appear or how they fit together globally.
Typical examples include:
- The projection $\pi : \mathbb{R} \times {0,1} \to \mathbb{R}$ given by $\pi(x,i)=x$; each point of $\mathbb{R}$ has a neighbourhood evenly covered by two disjoint copies of $\mathbb{R}$.
- The exponential map $\exp : \mathbb{C} \to \mathbb{C}^\times$ (complex plane minus zero) is a local homeomorphism but not a covering map because the fibres are infinite and the map fails to be proper.
- The inclusion of an open subset $U\subseteq X$ into $X$ is a local homeomorphism (trivially, with $U$ itself as the neighbourhood).
Local homeomorphisms appear prominently in differential geometry (as local diffeomorphisms), complex analysis (as local biholomorphisms), and the theory of sheaves, where they serve as the basis for the étale topology.
Etymology / Origin
The term combines homeomorphism—from the Greek roots “homo” (same) and “morphe” (form), denoting a bijective continuous map with continuous inverse—and the prefix local, indicating that the homeomorphic property holds only on neighbourhoods of points rather than globally. The concept was formalized in the early to mid‑20th century in the development of algebraic topology and covering space theory.
Characteristics
| Property | Description |
|---|---|
| Continuity | By definition, the map $p$ is continuous. |
| Local bijectivity | For each $b\in B$ there exists a neighbourhood $U$ such that $p |
| Open mapping | A local homeomorphism is an open map; images of open sets are open in $B$. |
| Closedness of fibres | Fibres $p^{-1}(b)$ are discrete (each point of a fibre has a neighbourhood mapping homeomorphically onto a neighbourhood of $b$). |
| Lifting of paths | Any path in $B$ starting at a point $b$ can be locally lifted to a path in $E$ beginning at a chosen point of the fibre over $b$. Global lifting may fail without additional hypotheses (e.g., covering maps). |
| Preservation of local properties | Local topological properties (local connectedness, local compactness, etc.) are transferred between $E$ and $B$ via a local homeomorphism. |
| Compatibility with differentiable structure | If $p$ is a smooth map between manifolds and its differential is everywhere invertible, then $p$ is a local diffeomorphism, a smooth analogue of a local homeomorphism. |
Related Topics
- Covering map – a surjective local homeomorphism with additional global regularity conditions.
- Local diffeomorphism – the differentiable counterpart on smooth manifolds.
- Étale map – in algebraic geometry, a morphism that is formally étale; topologically it behaves like a local homeomorphism.
- Sheaf theory – local homeomorphisms underpin the definition of sheaves on the étale site.
- Fiber bundle – a map with locally trivial fibres; every covering map is a fiber bundle with discrete fibre.
- Fundamental group – the theory of covering spaces (a special case of local homeomorphisms) provides a classification of subgroups of the fundamental group.
References
- Hatcher, A. Algebraic Topology. Cambridge University Press, 2002.
- Munkres, J.R. Topology. 2nd ed., Prentice Hall, 2000.
- Warner, F.W. Foundations of Differentiable Manifolds and Lie Groups. Springer, 1983.
(The above references discuss local homeomorphisms in the contexts of topology, covering spaces, and differential geometry.)