micromass said:Hi lavster!
I thought it could be useful if I gave some motivation about the notion of "compactness". In analysis, there are a few important theorems, one of them is the extreme value theorem.
As you probably know, this states that
This theorem is extremely useful as it can be used to show Rolle's theorem, Taylor's theorem and several others. So mathematicians began to think whether they could probably change the domain [a,b] to a more general one (why? because the theorem could have useful generalization to multivariate analysis!). And this was indeed possible! We can generalize the theorem as follows:
Still, mathematicians were not happy. Indeed, several developments in mathematics and physics lead them to consider metric spaces and topological spaces. These are concepts in which notions such as continuousness and convergence still make sense.
The question is whether we could generalize the extreme value theorem to the setting of metric spaces and topological spaces. This is indeed possible and it leads us to the notion of compactness. In metric spaces, a space is compact if every sequence has a convergent subsequence, or equivalently, if every open cover has a finite subcover. This is the correct generalization, since we have
Furthermore, this notion of compactness specializes to closed and boundedness for subsets of [tex]\mathbb{R}^n[/tex] (this is the Heine-Borel theorem). Thus this definition is really what we're looking for!
For topological spaces, the situation becomes slightly more complicated as the statements "every sequence has a convergent subsequence" and "every cover has a finite subcover" are no longer equivalent. It was discovered, however, that the latter was the proper generalization. The sequence-statement was less important, but still got the name sequential compactness.
Also note that the history of compactness was a very long one. Right now, we can give relatively easy definitions of compactness, but the original definitions were quite complicated!
A compact space is a mathematical concept that refers to a topological space in which every open cover has a finite subcover. In simpler terms, it is a space in which every open set can be covered by a finite number of smaller open sets.
In general, compactness is closely related to continuity. A function is continuous if and only if the preimage of any open set is an open set. In a compact space, every open set can be covered by a finite number of smaller open sets, so the preimage of any open set is also open. This means that continuous functions on compact spaces have nice properties and can be easily studied.
Some common examples of compact spaces include closed intervals in the real line, finite sets, and the Cantor set. Other examples include compact subsets of Euclidean spaces, compact manifolds, and compact Hausdorff spaces.
Compactness and connectedness are both important concepts in topology, but they are not the same. A space is compact if every open cover has a finite subcover, while a space is connected if it cannot be partitioned into two non-empty disjoint open sets. In other words, compactness deals with the size of open covers, while connectedness deals with the division of a space into smaller parts.
Understanding compact spaces is important in many areas of mathematics, including analysis, geometry, and topology. Compact spaces have properties that make them easier to study and can help simplify complicated problems. They also have applications in physics, engineering, and computer science, making them a valuable concept to understand for any scientist.