Question

define compact space

Answer 1

compactness is a property that generalizes the notion of a subset of a eucledian  space being closed and bounded.

Answer 2

In mathematics, and more specifically in general topology,compactness is a property that generalizes the notion of a subset of Euclidean space being closed (that is, containing all its limit points) and bounded (that is, having all its points lie within some fixed distance of each other). Examples include aclosed interval, a rectangle, or even a finite set of points. This notion is extended to more general topological spaces in various ways.