On some topological structures of topological monoids

Let G be a topological monoid, meaning that it is both a monoid and a topological space with continuous multiplication. This paper focuses on points in G that do not possess compact neighborhoods. In the context of topological groups, if the identity element, denoted e, has a compact neighborhood, then the space is locally compact. However, in the context of topological monoids, we construct an example where the identity element has a compact neighborhood while the space is not locally compact. Elements x and y in G are called mutually inverse if their products xy and yx equal e. We first investigate the lack of compact neighborhoods in topological monoids and confirm that results applicable to topological groups also hold true for topological monoids in the case of mutually inverse elements. Next, we introduce the concept of a strictly mutually inverse pair (y, x), where yx = e but xy does not necessarily equal e. In this case, we demonstrate that specific relationships between x, y, and their neighborhoods are relevant when considering left and right inverses within this context.