The topological group G is a group that is also a topological space satisfying the T1 Axiom that is, there is a topology on the elements of G and all sets of a finite number .
Given :
A topological group G is a group that is also a topological space satisfying the T1 axiom, such that the map of G * G into G sending x * y into x⋅y, and the map of G into G sending x into x^−1, are continuous maps.
When we say the map of G * G into G defined as (x, y) 7 → x·y is continuous, we use the product topology on G * G. Since inversion maps G to G, the topology on G is used both in the domain and co - domain.
Every group G is a topological group. We just equip G with the discrete
topology. Continuity of the binary operation follows at (x, y) ∈ G * G by taking the open set O = { x · y } (the “δ set” ) for any given open set in G * G containing (x, y) (the “ε set”).
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