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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.
A topological group, G, is a topological space that is also a group such that the group operation (in this case product): ⋅ : G × G → G, (x, y) ↦ xy. and the inversion map: −1 : G → G, x ↦ x −1. are continuous. [note 1] Here G × G is viewed as a topological space with the product topology.
The strong topological invariants of a many-band system in dimensions can be labeled by the elements of the -th homotopy group of the symmetric space. These groups are displayed in this table, called the periodic table of topological insulators:
Topological groups (2 C, 56 P) U. ... Generalized space; Geometric topology (object) Graph (topology) ... Topological vector space; List of topologies;
The empty set (considered as a topological space) is the initial object of Top; any singleton topological space is a terminal object. There are thus no zero objects in Top. The product in Top is given by the product topology on the Cartesian product. The coproduct is given by the disjoint union of topological spaces.
Topological Homogeneity. A space X is (topologically) homogeneous if for every x and y in X there is a homeomorphism : such that () =. Intuitively speaking, this means that the space looks the same at every point. All topological groups are homogeneous. Finitely generated or Alexandrov.
The term topology was introduced by Johann Benedict Listing in the 19th century, although it was not until the first decades of the 20th century that the idea of a topological space was developed. This is a list of topology topics. See also: Topology glossary; List of topologies; List of general topology topics; List of geometric topology topics
List of NGC objects. List of NGC objects (1–1000) List of NGC objects (1001–2000) List of NGC objects (2001–3000) List of NGC objects (3001–4000) List of NGC objects (4001–5000) List of NGC objects (5001–6000) List of NGC objects (6001–7000) List of NGC objects (7001–7840) List of IC objects; List of Messier objects; List of ...