Abstract

Topological groups are objects that combine two separate structures the structure of a topological space and the algebraic structure of a group?linked by the requirement that the group operations are continuous with respect to the underlying topology. Many of the natural infinite groups one encounters in mathematics are in fact topological. With regard to this definition, it is easy to see that oddly enough, if a set is not open, it does not mean that it is necessarily closed. It is possible for a set to be neither closed nor open, or both closed and open at the same time. In fact, we are guaranteed two such sets in the definition of a topology ¥ó. Both X and the empty set are guaranteed to be open, and because they are each other¡¯s complements, they are both guaranteed to be closed as well.
