25/55a Problem Set B - Clarifications

Several notes concerning the current problem set:

(first message)

Problem 2a. Yes, you must prove these are metrics, and that they are finite. Also note that an infinite sum is defined as the limit of the partial sums. That is, the sum from 1 to infinity of a_n is (by definition) the limit, as N goes to infinity, of the sum from 1 to N of a_n.

Problem 4a. Please note that a set can be closed but not open, open but not closed, both open and closed, or neither, and this definition is about whether certain sets are both open and closed. A map f from X to Y (where X and Y are any sets) is surjective iff for every element y of Y there exists an element x of X such that f(x)=y. A surjective map is also called onto. If you know what an image is, this definition is equivalent to: The image of f is Y.

Problem 4f. The definition of homeomorphism was incorrectly or imprecisely stated in class. A map f from X to Y (where X and Y are metric spaces) is a homeomorphism iff:

f is continuous
f is a 1-1 correspondence (also called a bijection), so that it has an inverse
This inverse is continous

(The definition from class merely said that f is 1-1.) If such a map exists, we say that X and Y are homeomorphic, or that X is homeomorphic to Y. (Note that X is homeomorphic to Y iff Y is homeomorphic to X. This is fairly straightforward to prove.)

Finally, although the last sentence says "If you're feeling energetic", it will still be graded as part of the problem set.

More news on prob set B

(second and last message)

I apologize that the length has been increasing as we comment on it. So, some good news: Problem 4c's comment, "Thus we see that...", is for your amusement and edification only; you don't have to prove it. It depends on the fact that a continuous function on a compact set is bounded, a fact we will prove in class soon.
I'd also like to note, by way of encouragement, that problem 4f's "feeling energetic" clause, which is indeed part of the problem, is easier than the other part of 4f (which is why we decided to grade it as part of the problem). Note also that we don't really expect many people to do the whole entire problem set. Also, this is the worst that 25ers will see in terms of problem sets.
In problem 2d, replace the comment "(Hard!)" with "(Hard!!!)". It's really hard. Don't stress about it if you don't get it.
I used the term "1-1 correspondence" in my last bulletin without defining it. Probably most of you know what this means, but for the record:
A function f from X to Y is a 1-1 correspondence iff: For every y in Y, there is exactly one x in X such that f(x)=y. (One can also say that a function is bijective iff it is both surjective and injective. I don't think we've defined injective, though. If the concepts are new to you you might want to look at Corwin-Szczarba, chapter 1, which reviews lots of concepts at the base of higher math, the sort of stuff that people like me tend to forget that we haven't defined.)

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