Quick convention question

[StephenPrivitera]

Is the closed interval [a,a] considered a legitimate notation for the set {a}? Would (a,a) denote the empty set?

 

[Ambitwistor]

I haven't actually seen anyone use that notation in such a degenerate case, but I don't see any reason why the usage you describe isn't legitimate.

 

[HallsofIvy]

[a,a] is used- usually in something like [a,b] where you want to consider the possibility that b= a.

If I came across reference to an interval like (a,a), I probably would interpret it as the empty set- although I would wonder why they picked a!

 

[StephenPrivitera]

A={x : f(x)<0 on [a,x]}
Is a in A? It is if f(x)<0 on [a,a]. And since [a,a] has only one number, it suffices to show that f(a)<0.

A={x : f(x)<0 on (a,x)}
Is a in A? It is if f(x)<0 on (a,a). But this is kind of nonsense. There is nothing in (a,a). You make the call.

So anyway, I just wanted to point this out to show why you might write (a,a) rather than {}.