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Quasitopological group


Abstract definition

A quasitopological group is a group equipped with a topology such that all the defining operations of a group (in the universal algebra sense) are separately continuous, i.e., each of the defining operations is continuous separately in each of its inputs.

Concrete definition

A quasitopological group is a set endowed with the following two structures:

  • The structure of a group, viz., an associative binary operation with identity element and inverses
  • The structure of a topological space

satisfying the following compatibility conditions:

Operation Arity Condition Comments
Multiplication or product 2 (so it's a map G \times G \to G) separately continuous as a map from G \times G to G. In other words, the group multiplication (g,h) \mapsto gh is continuous in each variable holding the other variable fixed. Separate continuity is weaker than joint continuity, which would mean continuity from G \times G equipped with the product topology to G. Note that the separate continuity condition is equivalent to the left multiplication map and right multiplication map by any element being self-homeomorphisms of G.
Identity element 0 (it's a constant element e \in G) no condition. As such, we may impose the condition that the map from a one-point space to G sending the point to the identity element is continuous, but this condition is vacuously true.
Inverse map 1 (so it's a map G \to G) continuous as a map from G to itself with the equipped topology. In other words, g \mapsto g^{-1} is continuous. Note that because the inverse map is its own inverse (see inverse map is involutive), this is equivalent to it being a self-homeomorphism of G.

Due to the fact that inverse map is involutive and hence that every group is naturally isomorphic to its opposite group via the inverse map, the continuity of the inverse map implies that simply assuming continuity of the multiplication in any one of its inputs implies separate continuity of multiplication. In particular, any left topological group where the inverse map is continuous is a quasitopological group, and similarly, any right topological group where the inverse map is continuous is a quasitopological group.

Relation with other structures

Stronger structures

Structure Meaning
topological group multiplication map is jointly continuous (i.e., continuous from the product topology) and inverse map is continuous
algebraic group see algebraic groups are quasitopological groups

Weaker structures

Structure Meaning
semitopological group multiplication map is separately continuous, no assumption about continuity of inverse map
left-topological group multiplication map is continuous in its right input, i.e., left multiplication maps are continuous
right-topological group multiplication map is continuous in its left input, i.e., right multiplication maps are continuous

Incomparable structures

  • Paratopological group: Here, the group multiplication is required to be jointly continuous, but we make no assumption about the continuity of the inverse map.