So here\'s the problem: Let X = ? ? [0,pi) be a subset of ?. Is X open in ?? Is

Question

So here's the problem:

Let X = ? ? [0,pi) be a subset of ?. Is X open in ?? Is X closed in ?? Prove your answer.

Can someone explain how to go about this problem? I understand that X is rational and is on a half open interval from 0 (which is closed) to pi (which is open). I understand that X is also a subset of ?. Then this is the part of the arguement that throws me off. It seems like it can be open, but then pi is an irrational number. If I argue that it's closed, well I'm too sure that's possible since 0 is rational and there are infinitely many rational number that can be found up to pi, and pi is open.

Explanation / Answer

I think in such type of questions topology will help you.

We can undoubtedly say X contains all rational numbers in the interval [0,pie].

Limit point: k is called a limit point of X,if every neighbourhood of k contains atleast one point belonging to X other than X.

Closed set: a set is closed if it contains all its limit points.

**pie is a limit point of X but is not contained in X.So X doesnot contain all its limit points therefore it is not closed.**

Interior point:a point j is said to be interior point if there exists a neighbourhood of j which is entirely contained in X.

Open set:If all the points of a set are interior points then the set is said to be open.

**Take any point in X,its any neighbourhood will contain some irration points too so it can be entirely contained in X as X contains rational points only.So X not Open**

Therefore X is neither OPEN nor CLOSED.

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