# Constructing a Bounded Closed set

1. Sep 24, 2009

### snipez90

1. The problem statement, all variables and given/known data
i) Construct a bounded closed subset of R (reals) with exactly three limit points
ii) Construct a bounded closed set E contained in R for which E' (set of limit points of E) is a countable set.

2. Relevant equations
Definition of limit point used: Let A be a subset of metric space X. Then b is a limit point of A if every neighborhood of b contains a point A different from b.

3. The attempt at a solution
All right so this seems pretty easy if you do it the lame way like I did. For i), you could just take the set containing 0 and 1/n for all natural numbers n, and this obviously has 0 as its only limit point. Have two other sets say, 1 with 1 + 1/n and 2009 with 2009 - 1/n. Clearly we have boundedness. Closed follows from intersection of sets which each contain their limit points.

It seems like we can extend the idea in i) to ii) as well (correct me if I'm wrong). However, is there a nicer way to construct these two sets?

2. Sep 24, 2009

### Dick

I think your 'lame' way is actually pretty nice.