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For example, a native version of the first isomorphism theorem is false for topological groups: if : is a morphism of topological groups (that is, a continuous homomorphism), it is not necessarily true that the induced homomorphism ~: / is an isomorphism of topological groups; it will be a bijective, continuous homomorphism, but it will not ...
The following is a list of named topologies or topological spaces, many of which are counterexamples in topology and related branches of mathematics. This is not a list of properties that a topology or topological space might possess; for that, see List of general topology topics and Topological property .
In mathematics, a topological group G is a group that is also a topological space such that the group multiplication G × G→G and the inverse operation G→G are continuous maps. Subcategories This category has the following 2 subcategories, out of 2 total.
This is a list of useful examples in general topology, a field of mathematics. Alexandrov topology; Cantor space; Co-kappa topology Cocountable topology; Cofinite topology; Compact-open topology; Compactification; Discrete topology; Double-pointed cofinite topology; Extended real number line; Finite topological space; Hawaiian earring; Hilbert cube
In the mathematical field of algebraic topology, the fundamental group of a topological space is the group of the equivalence classes under homotopy of the loops contained in the space. It records information about the basic shape, or holes, of the topological space. The fundamental group is the first and simplest homotopy group.
Toggle Properties and examples subsection. 1.1 Topology. 2 ... the homeomorphism group of a topological space is the group consisting of all homeomorphisms from the ...
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.
As in the case of topological groups, many deeper results require the point space to be (locally) compact and connected. This generalizes the observation that the line joining two distinct points in the Euclidean plane depends continuously on the pair of points and the intersection point of two lines is a continuous function of these lines.