Functions attaining max value => domain compact

[Mosis]

Homework Statement

Prove that if every continuous real-valued function on a set X attains a maximum value, then X must be compact.

Homework Equations

None really.

The Attempt at a Solution

None really, not sure where to start. I know that if a space is compact, every function attains it's minimum and maximum values, and I was thinking that I could show that if the space is not compact, then there is some continuous real-valued function that cannot attain a max value. I'm not really sure where to begin, though. Suggestions?

 

[mighty2000]

Hi there,

You're on the right track with your thinking! To prove this statement, we can use the contrapositive approach. That is, we will show that if X is not compact, then there exists a continuous real-valued function on X that does not attain a maximum value.

First, we need to define what it means for a space to be compact. A space X is compact if and only if every open cover of X has a finite subcover. In simpler terms, this means that every open cover of X (a collection of open sets that covers X) can be reduced to a finite number of open sets that still cover X.

Now, let's assume that X is not compact. This means that there exists an open cover C of X that cannot be reduced to a finite subcover. Let's call this open cover C = {Uα}, where α is an index set. Since C cannot be reduced to a finite subcover, it must be an infinite open cover.

Next, we will construct a continuous real-valued function f on X that does not attain a maximum value. To do this, we will use the fact that X is not compact, which means that there exists a sequence of points {x_n} in X that has no convergent subsequence. This means that for every x in X, there exists an open set U_x containing x such that U_x does not contain infinitely many points of the sequence {x_n}.

Now, let f(x) = n for x in U_x. In other words, f(x) takes on the value of the index n for every x in U_x. Since U_x does not contain infinitely many points of the sequence {x_n}, f(x) does not attain a maximum value on U_x. Furthermore, since this is true for every x in X, we can conclude that f(x) does not attain a maximum value on X.

Therefore, we have shown that if X is not compact, there exists a continuous real-valued function on X that does not attain a maximum value. This proves the contrapositive of the original statement, and therefore, the original statement must be true - that if every continuous real-valued function on a set X attains a maximum value, then X must be compact.

I hope this helps! Let me know if you have any questions or if you need any further clarification. Good luck with your proof!