# Topology and Compactness

1. Nov 12, 2004

### cogito²

My question comes from homework from a section on the Tychonoff Theorem. This is the question:

Now I have an idea about how to go about this. I know that $$Q$$ is compact since $$I = [0,1]$$ is and the Tychonoff Theorem states that the product of compact spaces is compact. I then know that $$f(Q) \subset \mathbb{R}$$ must be compact, since $$f$$ is continuous. I then know that it is closed and bounded meaning that there is an interval $$[-M,M]$$ such that $$f(Q) \subset [-M,M]$$. I can then cut that interval into a finite number of (non-disjoint) open sets of length $$\epsilon$$.

I would then look at their inverse images and see that they are open sets covering $$Q$$. I know that those open sets are unions of base sets that are products of sets of which only a finite number are not equal to $$I$$ (because this is the product topology). The problem for me is here is that the open sets cannot necessarilly be written as products (of sets) so I don't see how I can ignore all but a finite number of coordinates. What I would like is for the open sets to be a finite union of base sets. If that were true than I could find a finite number of coordinates that it belongs to but if not than I don't see how I can define it.

Beyond that I would like to define my function $$g$$ as a sum of functions $$g_i$$ each defined on the partitions of $$[-M,M]$$. I would be tempted to just define the functions to have a value of part of the partition but then that would result in a step function that wouldn't be continuous anyway. If I could come up with a set of continuous functions $$\{g_i\}$$ then my plan is to use the Partition of Unity to combine them into one function $$x$$ that would be the result of the problem.

So I'm pretty stumped at the moment. I just don't see how to simplify it. Hopefully my explanation is understandable. Any help would be greatly appreciated.