Welltopologized

Welltopologized is a term used in abstract algebra, specifically within the study of topological groups. A topological group G is said to be welltopologized if its topology satisfies certain desirable properties that make it behave well with respect to its group operations.

A common definition of a welltopologized group is that it is a topological group where the set of neighborhoods of the identity element forms a base for the topology at the identity. This means that for any open set containing the identity element, there exists a neighborhood of the identity that is contained within that open set. This property ensures that the topology is "rich" enough to capture the group structure effectively.

Alternatively, a topological group can be considered welltopologized if it is Hausdorff and has a basis of neighborhoods of the identity that are symmetric, meaning that if U is a neighborhood of the identity, then its inverse U⁻¹ is also a neighborhood of the identity. This symmetric neighborhood condition is often inherited from the underlying topological space.

The concept of welltopologized groups is important because many standard topological groups encountered in mathematics, such as Lie groups or locally compact groups, are welltopologized. This property simplifies many proofs and allows for the application of various powerful theorems in topological group theory. In essence, being welltopologized ensures that the topology is compatible with the group structure in a way that avoids pathological behavior.