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Some authors use the term Alexander trick for the statement that every homeomorphism of can be extended to a homeomorphism of the entire ball .. However, this is much easier to prove than the result discussed above: it is called radial extension (or coning) and is also true piecewise-linearly, but not smoothly.
Any closed topological n-ball is homeomorphic to the closed n-cube [0, 1] n. An n-ball is homeomorphic to an m-ball if and only if n = m. The homeomorphisms between an open n-ball B and R n can be classified in two classes, that can be identified with the two possible topological orientations of B. A topological n-ball need not be smooth; if it ...
As trivial examples, note that attaching a 0-handle is just taking a disjoint union with a ball, and that attaching an n-handle to (,) is gluing in a ball along any sphere component of . Morse theory was used by Thom and Milnor to prove that every manifold (with or without boundary) is a handlebody, meaning that it has an expression as a union ...
In mathematics and more specifically in topology, a homeomorphism (from Greek roots meaning "similar shape", named by Henri Poincaré), [2] [3] also called topological isomorphism, or bicontinuous function, is a bijective and continuous function between topological spaces that has a continuous inverse function.
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In the argument below denotes an infinite-dimensional separable Fréchet space and the relation of topological equivalence (existence of homeomorphism). A starting point of the proof of the Anderson–Kadec theorem is Kadec's proof that any infinite-dimensional separable Banach space is homeomorphic to R N . {\displaystyle \mathbb {R} ^{\mathbb ...
Together they yield a homeomorphism F 2 of the closure of the interior of Γ 2 onto the closure of the interior of Δ 2; F 2 extends F 1. Continuing in this way produces polygonal curves Γ n and triangles Δ n with a homomeomorphism F n between the closures of their interiors; F n extends F n – 1.
The connected sum of two n-manifolds is defined by removing an open ball from each manifold and taking the quotient of the disjoint union of the resulting manifolds with boundary, with the quotient taken with regards to a homeomorphism between the boundary spheres of the removed balls.