Characterizations of The Category of Topological Spaces - Definition Via Convergence

Definition Via Convergence

The category of topological spaces can also be defined via a convergence relation between filters on X and points of x. This definition demonstrates that convergence of filters can be viewed as a fundamental topological notion. A topology in the usual sense can be recovered by declaring a set A to be closed if, whenever F is a filter on A, then A contains all points to which F converges.

Similarly, Top can be described via net convergence. As for filters, this definition shows that convergence of nets can be viewed as a fundamental topological notion. A topology in the usual sense can be recovered by declaring a set A to be closed if, whenever (x_α) is a net on A, then A contains all points to which (x_α) converges.