On group uniformities on the square of a space and extending pseudometrics

Abstract

We give some conditions under which, for a given pair ( d 1 , d 2 ) of continuous pseudometrics respectively on X and X 3 , there exists a continuous semi-norm N on the free topological group F ( X ) such that N ( x · y −1 ) = d 1 ( x, y ) and N ( x · y · t −1 · z −1 ) ≥ d 2 (( x, y ), ( z, t )) for all x, y, z, t ∈ X . The “extension” results are applied to characterise thin subsets of free topological groups and obtain some relationships between natural uniformities on X 2 and those induced by the group uniformities * V, V * and * V * of F ( X ).

Locations

• Bulletin of the Australian Mathematical Society - View - PDF