cogito²
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My question comes from homework from a section on the Tychonoff Theorem. This is the question:
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.
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.Problem said:Let Q = I^A be a cube, and let f be a continuous real-valued function on Q. Prove, given \epsilon > 0, there is a continuous real-valued function g on Q such that |f - g| < \epsilon and g is a function of only a finite number of coordinates. [Hint: Cover the range of f by a finite number of intervals of length \epsilon 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 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.