Convergence in topological spaces

This paper will show the inadequacy of sequences to define certain concepts in topological spaces as fundamental as the real numbers. It introduces a generalization of a sequence, called a net, and shows that with nets it is possible to overcome this inadequacy. The idea of a Cauchy net in the real numbers R is defined, and a Cauchy criterion for nets in R is proved. Then it is shown that subnets exist (corresponding to subsequences) and generalizations of the usual theorems on sequences are given. Basic topological concepts such as Hausdorff and compact spaces, continuous functions, and the closure operator are then shown to be definable in terms of convergence of nets. Finally, alternative methods of discussing convergence in topological spaces are given and it is shown that convergence in terms of them is equivalent to convergence in terms of nets