versus homeomorphisms Every homeomorphism is a local homeomorphism. But a local homeomorphism 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 local homeomorphism 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 local homeomorphisms. That is, if X is locally Euclidean of dimension n and f : Y → X is a local homeomorphism, then Y is locally Euclidean...

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...

{\displaystyle U} is an open subset of M {\displaystyle M} ) and a local homeomorphism. There are also generalizations to certain types of continuous maps...

many-to-one map. Note that it is not a covering map – while it is a local homeomorphism near the origin, it is not a covering map at rotations by 180 degrees...

(mathematical analysis) – Connected open subset of a topological space Local homeomorphism – Mathematical function revertible near each point Open map – A function...

{\displaystyle F} is a topological space E {\displaystyle E} together with a local homeomorphism π : E → X {\displaystyle \pi :E\to X} such that the sheaf of sections...

the exponential map A ↦ e i A {\displaystyle A\mapsto e^{iA}} is a local homeomorphism 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 local homeomorphism, because...

{\displaystyle X\subseteq Y} . In general topology, an embedding is a homeomorphism onto its image. More explicitly, an injective continuous map f : X →...

F-sigma set Closeness Neighborhood Continuity (topology) Homeomorphism Local homeomorphism Open and closed maps Embedding Germ Basis Subbasis Open cover...

for three-dimensional rigid bodies, formally named SO(3)) is not a local homeomorphism at every point, and thus at some points the rank (degrees of freedom)...

since the inclusion map U → X {\displaystyle U\to X} is an open local homeomorphism. Using Hilbert space microbundles, David Henderson showed in 1969...

. {\displaystyle X.} In particular, if f {\displaystyle f} is a local homeomorphism from X {\displaystyle X} to Y {\displaystyle Y} , each fiber of f...

} At most points of the Riemann sphere, this transformation is a local homeomorphism: it maps a small disk centered at any point in a one-to-one way into...

algebras, Lie theory works less well. The exponential map need not be a local homeomorphism (for example, in the diffeomorphism group of the circle, there are...

{\displaystyle f:X\to Z} , Z {\displaystyle Z} some topological space, is a local homeomorphism that is injective on A {\displaystyle A} , then f {\displaystyle...

→ G {\displaystyle \exp :\operatorname {Lie} (G)\to G} defines a local homeomorphism from a neighborhood of the zero vector to the neighborhood of the...

LH whose objects are topological spaces and whose morphisms are local homeomorphisms is locally Cartesian closed, since LH/X is equivalent to the category...

{\displaystyle N} of the equilibrium u ∗ {\displaystyle u^{*}} and a homeomorphism h : N → R n {\displaystyle h\colon N\to \mathbb {R} ^{n}} , such that...

\varphi _{\beta }} are both homeomorphisms, the transition map τ α , β {\displaystyle \tau _{\alpha ,\beta }} is also a homeomorphism. One often desires more...

[clarification needed] Depending on the context, we can take this as local homeomorphism for the strong topology, over the complex numbers, or as an étale...

that φ is a homeomorphism onto its image; in the usage of some authors, this may instead mean that φ : U → Rn is itself a homeomorphism. The presence...

that P(Σ) is the whole upper half plane. On the other hand, P is a local homeomorphism, so a covering map. Since the upper half plane is simply connected...

(respectively, smooth manifold). Since two homeomorphisms h : U → V and g : V → W compose to a homeomorphism from U to W, one asks that the pseudogroup...

maps are local homeomorphisms. Even more remarkable is that every quasiregular local homeomorphism Rn → Rn, where n ≥ 3, is a homeomorphism (this is a...