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4-manifold. In mathematics, a 4-manifold is a 4-dimensional topological manifold. A smooth 4-manifold is a 4-manifold with a smooth structure. In dimension four, in marked contrast with lower dimensions, topological and smooth manifolds are quite different. There exist some topological 4-manifolds which admit no smooth structure, and even if ...
The separation axioms are about the use of topological means to distinguish disjoint sets and distinct points. It's not enough for elements of a topological space to be distinct (that is, unequal); we may want them to be topologically distinguishable. Similarly, it's not enough for subsets of a topological space to be disjoint; we may want them ...
Pushout (category theory) In category theory, a branch of mathematics, a pushout (also called a fibered coproduct or fibered sum or cocartesian square or amalgamated sum) is the colimit of a diagram consisting of two morphisms f : Z → X and g : Z → Y with a common domain.
Kirby calculus. In mathematics, the Kirby calculus in geometric topology, named after Robion Kirby, is a method for modifying framed links in the 3-sphere using a finite set of moves, the Kirby moves. Using four-dimensional Cerf theory, he proved that if M and N are 3-manifolds, resulting from Dehn surgery on framed links L and J respectively ...
A Postnikov system of a path-connected space is an inverse system of spaces. with a sequence of maps compatible with the inverse system such that. The map induces an isomorphism for every . Each map is a fibration, and so the fiber is an Eilenberg–MacLane space, . The first two conditions imply that is also a -space.
For a torus, the first Betti number is. b. 1. = 2 , which can be intuitively thought of as the number of circular "holes". Informally, the k th Betti number refers to the number of k -dimensional holes on a topological surface. A " k -dimensional hole " is a k -dimensional cycle that is not a boundary of a (k +1)-dimensional object.
Many examples of bounded convex polytopes can be found in the article "polyhedron".In the 2-dimensional case the full-dimensional examples are a half-plane, a strip between two parallel lines, an angle shape (the intersection of two non-parallel half-planes), a shape defined by a convex polygonal chain with two rays attached to its ends, and a convex polygon.
Definition 1. Informally, a space whose topology is determined by its compact subspaces, or equivalently in this case, by all continuous maps from arbitrary compact spaces. A topological space is called compactly-generated or a k-space if it satisfies any of the following equivalent conditions: [2][3][4] (1) The topology on.