Denseness

Surinder Pal Singh Kainth ()
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Surinder Pal Singh Kainth: Panjab University, Department of Mathematics

Chapter Chapter 8 in A Comprehensive Textbook on Metric Spaces, 2023, pp 213-241 from Springer

Abstract: Abstract Uncountable spaces are often difficult to handle, as one can’t ‘list up’ all the elements. The situation is much better when a metric space contains a countable dense subset, as that can approximate all its elements and thus is a good representative of the space itself (e.g. $$\mathbb {Q}$$ in $$\mathbb {R}$$ ). Such spaces are known as separable spaces. In this chapter, we discuss some notions emanating from denseness such as separability, perfect sets, Baire category, and equicontinuity. We start with a section on separable spaces, which also presents some standard Polish spaces, along with the relationship of separability with different types of bases such as the topological bases and the Schauder bases. Then we introduce perfect sets and discuss the Cantor–Bendixon theorem. It is followed by the Baire Category Theorem, along with a variety of applications. We wind up this chapter with equicontinuity and related results on the compactness of C[a, b].

Date: 2023
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DOI: 10.1007/978-981-99-2738-8_8

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