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Cardinality

In: A Comprehensive Textbook on Metric Spaces

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  • Surinder Pal Singh Kainth

    (Panjab University, Department of Mathematics)

Abstract

Can you write a given set as a sequence? In particular, can you write all real numbers together as a sequence? Such questions lead to the notions of countable and uncountable sets, and cardinality in general. This chapter discusses these ideas, along with their applications. It starts with countable and uncountable sets, followed by two special sections entitled ‘Some Applications to Topology’ and ‘The Set of Discontinuities’. The latter includes the case of monotone functions along with the general case, which asserts that the set of discontinuities of a function between metric spaces is a countable union of closed sets. Finally, there is a section on cardinality which provides a glimpse into cardinal arithmetic.

Suggested Citation

  • Surinder Pal Singh Kainth, 2023. "Cardinality," Springer Books, in: A Comprehensive Textbook on Metric Spaces, chapter 0, pages 183-211, Springer.
    • Handle: RePEc:spr:sprchp:978-981-99-2738-8_7
    DOI: 10.1007/978-981-99-2738-8_7
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