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
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versus homeomorphisms Every homeomorphism is a localhomeomorphism. But a localhomeomorphism is a homeomorphism if and only if it is bijective. A local homeomorphism...
{\displaystyle X} and Y {\displaystyle Y} are homeomorphic. A self-homeomorphism is a homeomorphism from a topological space onto itself. Being "homeomorphic"...
there exists a homeomorphism h : E → E ′ {\displaystyle h:E\rightarrow E'} , such that the diagram commutes. If such a homeomorphism exists, then one...
a local diffeomorphism between two manifolds exists then their dimensions must be equal. Every local diffeomorphism is also a localhomeomorphism and...
U\times F} is a homeomorphism. The set of all { ( U i , φ i ) } {\displaystyle \left\{\left(U_{i},\,\varphi _{i}\right)\right\}} is called a local trivialization...
Euclidean is preserved by localhomeomorphisms. That is, if X is locally Euclidean of dimension n and f : Y → X is a localhomeomorphism, then Y is locally Euclidean...
for three-dimensional rigid bodies, formally named SO(3)) is not a localhomeomorphism at every point, and thus at some points the rank (degrees of freedom)...
{\displaystyle U} is an open subset of M {\displaystyle M} ) and a localhomeomorphism. There are also generalizations to certain types of continuous maps...
codomain is essential. Every homeomorphism is open, closed, and continuous. In fact, a bijective continuous map is a homeomorphism if and only if it is open...
the exponential map A ↦ e i A {\displaystyle A\mapsto e^{iA}} is a localhomeomorphism from the space of self-adjoint complex matrices to U(n). The space...
sends a local orientation at p to p. It is clear that every point of M has precisely two preimages under π. In fact, π is even a localhomeomorphism, because...
algebras, Lie theory works less well. The exponential map need not be a localhomeomorphism (for example, in the diffeomorphism group of the circle, there are...
{\displaystyle X\subseteq Y} . In general topology, an embedding is a homeomorphism onto its image. More explicitly, an injective continuous map f : X →...
(mathematical analysis) – Connected open subset of a topological space Localhomeomorphism – Mathematical function revertible near each point Open map – A function...
[clarification needed] Depending on the context, we can take this as localhomeomorphism for the strong topology, over the complex numbers, or as an étale...
{\displaystyle f:X\to Z} , Z {\displaystyle Z} some topological space, is a localhomeomorphism that is injective on A {\displaystyle A} , then f {\displaystyle...
since the inclusion map U → X {\displaystyle U\to X} is an open localhomeomorphism. Using Hilbert space microbundles, David Henderson showed in 1969...
(mathematics) neighbourhood (mathematics) Continuity (topology) HomeomorphismLocalhomeomorphism Open and closed maps Germ (mathematics) Base (topology), subbase...
maps are localhomeomorphisms. Even more remarkable is that every quasiregular localhomeomorphism Rn → Rn, where n ≥ 3, is a homeomorphism (this is a...
properties these problems do rely on. From this need arises the notion of homeomorphism. The impossibility of crossing each bridge just once applies to any...
normal etc... Locally metrizable Given some notion of equivalence (e.g., homeomorphism, diffeomorphism, isometry) between topological spaces, two spaces are...
spherical polyhedron. Further, because a covering map is a localhomeomorphism (in this case a local isometry), both the spherical and the corresponding projective...
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