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 (xα) is a net on A, then A contains all points to which (xα) converges.
Read more about this topic: Characterizations Of The Category Of Topological Spaces
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