Topological Spaces

Zigang Pan

Chapter Chapter 3 in Measure-Theoretic Calculus in Abstract Spaces, 2023, pp 41-77 from Springer

Abstract: Abstract Topological space is a set X together with a collection of subsets of X, which we call open sets. Once we have the notion of open sets, one can talk about closed sets, the closure of a set, the interior of a set, the boundary of a set, etc. The notions of limit and continuity of a function is intrinsically linked to topological spaces. We touch on the notions of basis of a topology, the countability of the topological space, the connectedness, the separation axioms for topological spaces. The category theory is introduced here and will have significance in Chap. 4 . We present the Urysohn’s Lemma and tiesze Extension Theorem that specifies that certain continuous functions exists for normal topological spaces. One key concept that we present is the notion of net. This is a generalized notion for sequences, and allows us to define integration for both Lebesgue and Riemann integrals.

Date: 2023
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DOI: 10.1007/978-3-031-21912-2_3

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