ABSTRACT
The notion of compactness is perhaps the least intuitive but plays a central role in Analysis. It was Frechet who coined the term “compact" in his 1906 doctoral thesis and gave the definition for what we now know as countable and sequential compactness. However, Alexandroff and Urysohn get credit for defining open cover compactness.
This chapter deals with compactness of metric spaces and its consequences. We begin with defining the basic intuitive notion of open covering which leads to the definition of covering compactness. Sufficient conditions for compactness and general properties of compact sets are discussed. Sequential compactness has been introduced as a generalization of Bolzano and Weirstrass theorem. The chapter is conceded with characterization theorem for compactness including Lebesue covering lemma.
