(x,x] = {x}

Fredrik

If you define (b,a] as "the interval from b to a, with b not included and a included", then (x,x] is neither empty nor non-empty, it's just nonsense. (A set can't both contain x and not contain x). If you define (b,a] as "the set of all t such that b<t≤a", then (x,x] is empty. (I prefer the latter definition because it makes sense for all a and b, and doesn't require us to have defined "interval" in advance).

I don't understand your concern at all. n is an index that labels the sets in the family of sets that we're taking the intersection of. If the definitions of the sets in the family hadn't involved some algebraic operations, we could have used any set as the index set. In this particular case, where the sets are (x-1/n,x], the index set can be any subset of the real numbers that doesn't include 0.

SW VandeCarr

I'm sorry. Why do you say that (3,2] doesn't include 2? As a half closed interval doesn't this notation correspond to [tex]3> x \geq 2[/tex]? Please explain why you are saying 2 is not included in the interval when every reference I've checked says it is. I'll just post one.

http://mathworld.wolfram.com/Half-ClosedInterval.html

Fredrik

No, (3,2] is the set of all real numbers x such that 3<x≤2. There are no such real numbers. Therefore, (3,2]=∅.

SW VandeCarr

OK. But that's not the inequality I wrote in my previous posts. However for (a,b] I do see your point. If I write [tex]a < x \leq b[/tex] then the corresponding set has no least element and therefore cannot be well ordered. Thanks.