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A homeomorphism is a special case of a homotopy equivalence, in which g ∘ f is equal to the identity map id X (not only homotopic to it), and f ∘ g is equal to id Y. [ 7 ] : 0:53:00 Therefore, if X and Y are homeomorphic then they are homotopy-equivalent, but the opposite is not true.
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.
A continuous map between two topological spaces induces a group homomorphism between the associated homotopy groups. In particular, if the map is a continuous bijection (a homeomorphism ), so that the two spaces have the same topology, then their i -th homotopy groups are isomorphic for all i .
The homotopy group functors assign to each path-connected topological space the group () of homotopy classes of continuous maps . Another construction on a space X {\displaystyle X} is the group of all self-homeomorphisms X → X {\displaystyle X\to X} , denoted H o m e o ( X ) . {\displaystyle {\rm {Homeo}}(X).}
A key concept in defining simplicial homology is the notion of an orientation of a simplex. By definition, an orientation of a k-simplex is given by an ordering of the vertices, written as (v 0,...,v k), with the rule that two orderings define the same orientation if and only if they differ by an even permutation.
As with other sets of maps between topological spaces, the homeomorphism group can be given a topology, such as the compact-open topology.In the case of regular, locally compact spaces the group multiplication is then continuous.
A homotopy between two paths. Paths and loops are central subjects of study in the branch of algebraic topology called homotopy theory.A homotopy of paths makes precise the notion of continuously deforming a path while keeping its endpoints fixed.
A group homomorphism between topological groups is continuous if and only if it is continuous at some point. [4] An isomorphism of topological groups is a group isomorphism that is also a homeomorphism of the underlying topological spaces. This is stronger than simply requiring a continuous group isomorphism—the inverse must also be continuous.