# Question about the set (a,a]

If S=[a,a) Can such a set exist? It implies that a is in S and not in S, which doesn't make sense, but it seems a problem I'm trying to do requires it to be considered empty.

The question is:

Let In = [an,bn) where

In+1 < In for all natural numbers n. [< denotes subset]

Give an example of those In for which the intersection of In (for all n) is empty.

I can't see any other way to construct an empty set.

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Hurkyl
Staff Emeritus
Gold Member
(a, a] would be the set of all x such that a < x <= a.

However, I think the question intends you to limit yourself to intervals whose endpoints are distinct.

Recall that when you take a nested intersection of nonempty closed sets, you never get the empty set. Since you're searching for behavior that is not demonstrated by closed sets, I would suggest focusing at the open end of your intervals, where they resemble closed sets the least.

Hi. Thanks for the response. Based on your definition [a,a) is indeed empty. Although I'm not sure why you think the question requires the endpoints to be distinct.

If every set In+1 is a subset of In, then the only way the intersection of In for all n is empty is if one of those sets is empty. To my eyes (which admittedly are stupid at times), the only set of the form [a,b) that is empty is when a=b (or b<a)

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I just learned that the set [a,a) is not a proper subset of itself. So if In=[a,a), In+1 can't satisfy In+1 < In.

Now I have no idea what's going on. If no In can be empty, how can the intersection of all In be empty in any case! HELP