Quotients of Raikov-complete topological groups

Abstract

A topological group X is called Raikov-complete if the two sides uniformity on X, that is the supremum $\mathcal L\vee \mathcal R$ of the left uniformity ??? and the right uniformity $\mathcal R$ on X, is complete.It will be proved that the quotient $(X/G, (\mathcal L/G) \vee (\mathcal R/G))$ of an inframetrizable Raikov-complete topological group X with a neutral subgroup G is complete.(If G is normal (and therefore neutral), the completeness of $(X/G, (\mathcal L/G) \vee(\mathcal R/G))$ is equivalent to the Raikov-completeness of the quotient group $X/G$).The proof consist in an intricate lifting of $(\mathcal L/G)\vee(\mathcal R/G) $-Cauchy filters. Where it is possible, the results on topological groups will be derived from results on uniform spaces.