topological group
Noun
Meaning
topological group (plural topological groups)
(topology) A group which is also a topological space and whose group operations are continuous functions.
Examples
THEOREM: if G is a locally euclidean, connected, simply connected topological group of dimension n greater than one, then G contains a closed proper subgroup of positive dimension.
A generalization to functions g taking values in any topological group is also possible.
A normal subgroup of a topological group, acting on the group by translation action, is a quotient space in the senses of topology, abstract algebra, and group actions simultaneously.
It follows that if a topological group is T 0 ( Kolmogorov ) then it is already T 2 ( Hausdorff ), even T 3½ ( Tychonoff ).
However, if the unit group is endowed with the subspace topology as a subspace of R, it may not be a topological group, because inversion on R × need not be continuous with respect to the subspace topology.
A topological group G is Hausdorff if and only if the trivial one-element subgroup is closed in G. If G is not Hausdorff then one can obtain a Hausdorff group by passing to the quotient space G/K where K is the closure of the identity.