DataD

A subset  of a topological space  is compact 

if for every open cover of , there exists a finite subcover of .

즉, S의 open cover들에 대해서, S의 finite subcover(element가 유한한 open cover of S의 subset)가 존재하는 경우, S가 compact하다고 한다.

cover

A family  of nonempty subsets of  whose union contains the given set  (and which contains no duplicated subsets) is called a cover (or covering) of 

 For example of cover

there is only a single cover of , 

namely . 

However, there are five covers of , 

namely , , , , and .

minimal cover

A minimal cover is a cover for which removal of one member destroys the covering property.

Open cover

Example of open cover

- an open cover of the real line, with respect to the Euclidean topology, is the set of all open intervals , where .

- The set of all intervals , where , is an open cover of the open interval .

Extreme Value Theorem

If a function f(x) is continuous on a closed interval [a,b], 

then f(x) has both a maximum and a minimum on interval [a,b]. 

If f(x) has an extremum on an open interval (a,b), 

then the extremum occurs at a critical point. 

This theorem is sometimes also called the Weierstrass extreme value theorem.

Proofs

The standard proof of the first proceeds by noting that f is the continuous image of a  compact set on the interval [a,b], 

so it must itself be compact.

Since  is compact, it follows that the image f([a,b]) must also be compact.

compact set on the interval [a,b] : finite open cover로 covering되는 set on the interval [a,b]

f는 continuous image이기 때문에 f 또한 compact 해야한다.

f 가 compact 하기 때문에, finite open cover가 존재 한다. 즉, extreme 값이 해당 domain에 대한 image set에서 존재한다.