My question comes from homework from a section on the Tychonoff Theorem. This is the question:(adsbygoogle = window.adsbygoogle || []).push({});

Now I have an idea about how to go about this. I know that [tex]Q[/tex] is compact since [tex]I = [0,1][/tex] is and the Tychonoff Theorem states that the product of compact spaces is compact. I then know that [tex]f(Q) \subset \mathbb{R}[/tex] must be compact, since [tex]f[/tex] is continuous. I then know that it is closed and bounded meaning that there is an interval [tex][-M,M][/tex] such that [tex]f(Q) \subset [-M,M][/tex]. I can then cut that interval into a finite number of (non-disjoint) open sets of length [tex]\epsilon[/tex]. Problem said:Let [tex]Q = I^A[/tex] be a cube, and let [tex]f[/tex] be a continuous real-valued function on [tex]Q[/tex]. Prove, given [tex]\epsilon > 0[/tex], there is a continuous real-valued function [tex]g[/tex] on [tex]Q[/tex] such that [tex]|f - g| < \epsilon[/tex] and [tex]g[/tex] is a function of only a finite number of coordinates. [Hint: Cover the range of [tex]f[/tex] by a finite number of intervals of length [tex]\epsilon[/tex] and look at the inverse images of these intervals.]

I would then look at their inverse images and see that they are open sets covering [tex]Q[/tex]. 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 [tex]I[/tex] (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 [tex]g[/tex] as a sum of functions [tex]g_i[/tex] each defined on the partitions of [tex][-M,M][/tex]. 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 [tex]\{g_i\}[/tex] then my plan is to use the Partition of Unity to combine them into one function [tex]x[/tex] 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.

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# Homework Help: Topology and Compactness

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