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In topology, a branch of mathematics, a retraction is a continuous mapping from a topological space into a subspace that preserves the position of all points in that subspace. [1] The subspace is then called a retract of the original space. A deformation retraction is a mapping that captures the idea of continuously shrinking a space into a ...
In mathematics, specifically algebraic topology, the mapping cylinder [1] of a continuous function between topological spaces and is the quotient = (([,])) / where the denotes the disjoint union, and ~ is the equivalence relation generated by
a retraction if a right inverse of f exists, i.e. if there exists a morphism g : b → a with f ∘ g = 1 b. a section if a left inverse of f exists, i.e. if there exists a morphism g : b → a with g ∘ f = 1 a. Every retraction is an epimorphism, and every section is a monomorphism. Furthermore, the following three statements are equivalent:
Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though usually most classify up to homotopy equivalence. Although algebraic topology primarily uses algebra to study topological ...
In mathematics, especially homotopy theory, the homotopy fiber (sometimes called the mapping fiber) [1] is part of a construction that associates a fibration to an arbitrary continuous function of topological spaces:. It acts as a homotopy theoretic kernel of a mapping of topological spaces due to the fact it yields a long exact sequence of ...
In mathematics, in particular in homotopy theory within algebraic topology, the homotopy lifting property (also known as an instance of the right lifting property or the covering homotopy axiom) is a technical condition on a continuous function from a topological space E to another one, B.
In topology, the Vietoris–Rips complex, also called the Vietoris complex or Rips complex, is a way of forming a topological space from distances in a set of points. It is an abstract simplicial complex that can be defined from any metric space M and distance δ by forming a simplex for every finite set of points that has diameter at most δ.
In mathematics, and specifically in topology, a CW complex (also cellular complex or cell complex) is a topological space that is built by gluing together topological balls (so-called cells) of different dimensions in specific ways. It generalizes both manifolds and simplicial complexes and has particular significance for algebraic topology. [1]