TITLE: How to understand the concept of compact space
QUESTION [16 upvotes]: the definition of compact space is: A subset K of a metric space X is said to be compact if every open cover of K contains finite subcovers. What is the meaning of defining a space is "compact". I found the explanation on wikipedia : "In mathematics, more specifically general topology and metric topology, a compact space  is an abstract mathematical space in which, intuitively, whenever one takes an infinite number of "steps" in the space, eventually one must get arbitrarily close to some other point of the space. " I can not understand it, or at least I can not get this by its definition. Can anybody help me?

REPLY [15 votes]: Terry Tao has a nice explanation in the Princeton companion to maths. The article's also on his website: 
http://www.math.ucla.edu/~tao/preprints/compactness.pdf