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Convergence

  • Tej Bahadur SinghEmail author
Chapter

Abstract

We are familiar with the notion of convergence of sequences of numbers (real or complex) and its useful role in analysis. This can also be generalized to topological spaces. However, sequences in an arbitrary topological space are inadequate for certain purposes, as we shall see in Sect. 4.1. This problem is dealt with in Sect. 4.2 by means of “nets” which are generalizations of sequences. A considerably more versatile notion, “filters”, is treated in Sect. 4.3. It will be seen that uniqueness of “limits” (i.e., no sequence or net converges to more than one point) needs separation of points by disjoint open sets in the space. This condition is named after Felix Hausdorff and will be studied in Sect. 4.4 along with two other separation axioms.

Copyright information

© Springer Nature Singapore Pte Ltd. 2019

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of DelhiDelhiIndia

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