Search results
Results from the WOW.Com Content Network
Given a bijective function f between two topological spaces, the inverse function f −1 need not be continuous. A bijective continuous function with continuous inverse function is called a homeomorphism. If a continuous bijection has as its domain a compact space and its codomain is Hausdorff, then it is a homeomorphism.
In mathematics, particularly topology, the homeomorphism group of a topological space is the group consisting of all homeomorphisms from the space to itself with function composition as the group operation.
The real numbers form a topological group under addition. In mathematics, topological groups are the combination of groups and topological spaces, i.e. they are groups and topological spaces at the same time, such that the continuity condition for the group operations connects these two structures together and consequently they are not independent from each other.
A function from one space to another is continuous if the preimage of every open set is open. Continuum A space is called a continuum if it a compact, connected Hausdorff space. Contractible A space X is contractible if the identity map on X is homotopic to a constant map. Every contractible space is simply connected. Coproduct topology
A function or map from one topological space to another is called continuous if the inverse image of any open set is open. If the function maps the real numbers to the real numbers (both spaces with the standard topology), then this definition of continuous is equivalent to the definition of continuous in calculus.
Alternatively, these representations can be defined on the K-vector space W of all functions G → K. It is in this form that the regular representation is generalized to topological groups such as Lie groups. The specific definition in terms of W is as follows. Given a function f : G → K and an element g ∈ G,
In mathematics, a locally compact group is a topological group G for which the underlying topology is locally compact and Hausdorff.Locally compact groups are important because many examples of groups that arise throughout mathematics are locally compact and such groups have a natural measure called the Haar measure.
The one-point compactification of a cusped hyperbolic manifold has a canonical CW decomposition with only one 0-cell (the compactification point) called the Epstein–Penner Decomposition. Such cell decompositions are frequently called ideal polyhedral decompositions and are used in popular computer software, such as SnapPea.