Question

Property (3) given above is also called sequential compactness. So we would say that a set E⊆V is sequentially compact if every sequence in E has a subsequence that converges in E. There is an equivalent definition of sequential compactness that we consider below

Answer 1

Sequential compactness is a mathematical concept in topology that ensures each sequence in a set has a convergent subsequence within the same set, often associated with the Heine-Borel theorem.

Step-by-step explanation:

The concept of sequential compactness is a fundamental topic in topology, a branch of mathematics that deals with the properties of space that are preserved under continuous transformations. If a set E ⊆ V is said to be sequentially compact, it means that for every sequence of points in E, there exists a subsequence that converges to a point also within E. This is significant in understanding the behavior of sequences in metric spaces and has important implications in analysis and other areas of mathematics. An equivalent definition of sequential compactness involves the notion of covering and asserts that a subset of a topological space is sequentially compact if every open cover has a finite subcover, which relates to the Heine-Borel theorem.