Convergence in topological spaces

UNCG Author/Contributor (non-UNCG co-authors, if there are any, appear on document)
Dargan Frierson (Creator)
The University of North Carolina at Greensboro (UNCG )
Web Site:
E. E. Posey

Abstract: 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.

Additional Information

Language: English
Date: 1970
Topological spaces

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