Continuous mappings | General topology | Functions and mappings | Homeomorphisms

Local homeomorphism

In mathematics, more specifically topology, a local homeomorphism is a function between topological spaces that, intuitively, preserves local (though not necessarily global) structure. If is a local homeomorphism, is said to be an étale space over Local homeomorphisms are used in the study of sheaves. Typical examples of local homeomorphisms are covering maps. A topological space is locally homeomorphic to if every point of has a neighborhood that is homeomorphic to an open subset of For example, a manifold of dimension is locally homeomorphic to If there is a local homeomorphism from to then is locally homeomorphic to but the converse is not always true. For example, the two dimensional sphere, being a manifold, is locally homeomorphic to the plane but there is no local homeomorphism (Wikipedia).

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Invariance of domain | Topological space | Proper map | Complex analysis | Fundamental theorem of algebra | Homeomorphism | Derivative | Topology | Codomain | Continuous function | First-countable space | Locally connected space | Quotient space (topology) | Domain of a function | Inverse function theorem | Inclusion map | Topos | Baire space | Second-countable space | Formally étale morphism | Hausdorff space | Locally compact space | Disjoint union (topology) | Local diffeomorphism | Differentiable manifold | Completely metrizable space | Complex plane | Mathematics | Ramification (mathematics) | Function (mathematics) | Equivalence of categories | Sphere | Sheaf (mathematics) | Holomorphic function | Winding number | Manifold | Scheme (mathematics) | Equivalence relation | Fiber (mathematics) | Étale morphism | Covering space | Subspace topology | Function composition | Normal space | Restriction (mathematics) | Discrete space | Circle | Image (mathematics) | Open set