overlineVU

Overlinevu is an informal mathematical notation used in some texts to denote the topological closure of the product set VU, where V and U are subsets of a topological group or topological semigroup G. The product set VU consists of all elements vu with v in V and u in U. The closure of VU, written as overline{VU} or described as overlinevu in certain informal usages, is the smallest closed set in G that contains VU.

Not all authors recognize or standardize the term overlinevu; the more common practice is to write overline{VU} or cl(VU). The symbol overline indicates a closure operator, which is idempotent and extensive: overline{VU} contains VU, and overline{overline{VU}} = overline{VU}. This closure is a closed subset of G and is independent of the particular representation of V and U, depending only on the sets themselves and the topology of G.

In practice, overline{VU} coincides with VU if VU is already closed. If V and U are subgroups or subspaces with the ambient topology, overline{VU} describes how their product fills out space under the topology, which is especially relevant when V and U are not closed individually.

Examples appear in the study of topological groups, where one may examine the closure of the product of two subgroups or subspaces to understand how their combined structure behaves topologically. Because overlinevu is not a standardized term, readers should consult the specific text for the exact meaning and notation used. See also closure in topology, product of subsets, and subgroups in topological groups.