Here are some test questions and my responses, did I get these right? These are the ones that I'm least certain about.(adsbygoogle = window.adsbygoogle || []).push({});

- Show that in every metric space, finite sets are always closed.

Finite sets contain no limit points. Hence every finite set contains all if its limit points.- Suppose that [tex]\{ N_{a} \}[/tex] is a collection of disjoint neighborhoods in [tex]\mathbb{R}^{2}[/tex]. Show that the collection is at most countable.

My proof was a little long but it went roughly like this:

Because the neighborhoods are disjoint and because the rationals are dense in the reals it is possible to locate a point (a, b) where a and b are rational numbers in each neighborhood that is in no other neighborhood. So, we have a 1-1 correspondence between a subset of the countable set [tex] \mathbb{Q} \times \mathbb{Q}[/tex] and the neighborhoods in [tex]\{ N_{a} \}[/tex]. Hence, the set is at most countable. (Looking back, I can see I was thinking ofpairwisedisjoint, which we never defined... Rudin only mentions that two sets can be disjoint. http://en.wikipedia.org/wiki/Disjoint_sets" [Broken] makes a distinction between pairwise disjoint and sets that merely have a disjoint intersection. I think a variation of this proof might still work, so can I even hope for partial credit?? Maybe?)- Let p be a point in an arbitrary metric space X. Let [tex]A=\bigcap_{r \in \mathbb{R} >0} N_{r}(p)[/tex] the intersection of all neighborhoods of p. Prove that A={p}.

I said that a nested intersection of open sets could not be empty, and p is the only element contained in every set hence the intersection must be p.- For a metric space X and a function [tex]f:X \rightarrow \mathbb{R}[/tex] define Z(f) as the set of all x in X such that f(x) = 0. Show that if f is continuous, then Z(f) is closed.

I said that {0} is a closed set in [tex] \mathbb{R}[/tex] because it is finite. Hence [tex]f^{-1}(\{ 0 \})[/tex] must be a closed set in X because f is continuous.

If you can let me know if any of these seem right or wrong it be a big help! I'm freaking out about my grade.

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# Analysis Test Questions (did I get these?)

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