Math 440, Fall 2014, Assignment 10

The moving power of mathematical invention is not reasoning but the imagination.

- Augustus de Morgan

Carefully define the following terms, then give one example and one non-example of each:

1. Compact space (as we defined it in class, in terms of nets).
2. Open cover.
3. Compact space (as most books define it, in terms of open covers).
4. Locally compact space.

Carefully state the following theorems (you do not need to prove them):

1. Theorem concerning the equivalence of the two definitions of compactness.
2. Theorem concerning the forward image of a compact subspace under a continuous map.
3. Theorem concerning closed subspaces of compact spaces.
4. Theorem concerning compact subspaces of Hausdorff spaces.
5. Theorem concerning compactness of the unit interval.
6. Extreme Value Theorem.

Solve the following problems:

1. Problems 17A(1), 17B(1-4), 17G(1) and 17G(3).

--------------------End of assignment--------------------

Questions:

What is Theorem concerning the equivalence of the two definitions of compactness? Is it just that X is compact iff every open cover of X has a finite subcover since we defined compactness in class as X is compact if every net in X has a cluster point or something else?

Yes, that's right. We defined compact to mean that every net has a cluster point, while most books define it to mean that every open cover has a finite subcover. The theorem is that these two conditions are equivalent. - Steven.Jackson (talk) 23:18, 13 December 2014 (UTC)