The upper and Roelcke uniformities generated from the left and right uniformities of a topological group

For the left and right uniformities $\mathcal U_L,\mathcal U_R$ of a topological group:

• the upper uniformity is their join $\mathcal U_L\vee\mathcal U_R$, with basic entourages $L_U\cap R_V$;
• the Roelcke uniformity is their meet $\mathcal U_L\wedge\mathcal U_R$, with a base of composites $L_U\circ R_V$.

For the upper structure, intersections are the usual base for the least filter containing both input uniformities; inverse and square-root axioms follow by shrinking the left and right factors separately.

For the Roelcke structure, left and right relations commute: $L_U\circ R_V=R_V\circ L_U$, both saying that $y\in VxU$. Their inverses are again such composites, and if $U_0^{\,2}\subseteq U$ and $V_0^{\,2}\subseteq V$, then $(L_{U_0}\circ R_{V_0})^{\circ2}\subseteq L_U\circ R_V.$ Finite intersections are refined by intersecting the identity neighbourhoods. Hence both displayed bases satisfy Uniform space in the entourage formulation. The names and formulas are kept separate.