(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Let (X,d) be a metric space. Can a set E in X be both open and closed? Can a point in E be both isolated and an interior point?

2. Relevant equations

I've used the metric defined as d(x,y)=1 for [itex]x\ne y[/itex] and 0 if x=y (we used this in a previous problem). I also used the following definitions in baby Rudin:

2.18

(b) A point p is a limit point of the set E if every neighborhood of p contains another point q such that q is in E

(c) If p is in E and p is not a limit point of E, then p is called an isolated point of E

(d) E is closed if every limit point of E is a point of E

(e) A point p is an interior point of E if there is a neighborhood N of p such that N is a subset of E

(f) E is open if every point of E is an interior point of E

3. The attempt at a solution

For the first question, I said yes, that a set can both be open and closed. For example, any subset E of a metric space that uses the above metric is open (because a neighborhood of 1/2 around any point p, [itex]N_{1/2}(p)[/itex], is just that point which is a subset of E). Likewise, any subset is closed because there are no limit points to contain - the same neighborhood [itex]N_{1/2}(p)[/itex] only contains p and no other points, so p is not a limit point.

For the second question, I again said yes, again using the same metric as above. Since every subset E is open, then every point in E is an interior point. On the other hand, there are no limit points, and so every p is an isolated point as well.

I don't think this stuff is difficult, really, just a different way of looking at things (especially for somebody who started as a physics major). It's just a lot of definitions to remember, and I wanted to make sure that I understand the definitions properly.

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# Homework Help: Open and Closed Sets in a Metric Space

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