Incomplete infima of complete uniformities related to topological groups and topological vector spaces (Q1174823)

We consider uniformities in terms of vicinities. Completeness is meant in the sense that all Cauchy filters (or, equivalently, all Cauchy nets) converge. A topological group \(X\) is called complete if its left uniformity \({\mathfrak L}\) (or, equivalently its right uniformity \({\mathfrak R}\)) is complete. This terminology applies as usual to topological vector spaces, in which case the left and right uniformities coincide. If \({\mathfrak V}\), \({\mathfrak W}\) are two complete uniformities on a set \(X\) then either infimum \({\mathfrak V}\wedge {\mathfrak W}\) (i.e., the finest uniformity on \(X\) which is coarser than \({\mathfrak V}\) and \({\mathfrak W}\)) need not be complete. The purpose of this article is to exhibit examples for this phenomenon under various circumstances, starting in a general uniform setting and then proceeding to more restricted situations in topological groups and topological vector spaces.