Must “real number interval” be defined in such a way as to include the empty set as an interval.

Note that I am not asking whether the empty set is an interval, I am asking whether in the context of real analysis if anything is lost if "number interval" is defined in such a way that excludes either $(1)$ singleton sets or $(2)$ the empty set.

What motivates this question is the common sense idea that a number interval should contain numbers, possibly even two numbers.

The core of my question lies in the phrase: is anything lost?

P.S. Would Mathematics Meta be a better place to ask this?

• Meta isn't appropriate for this question; Meta is for questions about the site itself. I think you've gone to the right place, although you may get some pushback asking for clarifications / context to help formulate an answer. – Nick Peterson Jan 20 '17 at 20:35
• You would no longer have the intersection of two intervals being an interval, which could be problematic. – Morgan Rodgers Jan 20 '17 at 20:35
• The nice thing about a broader definition is that one can then attach a adjective to it to refer to a restricted class: "nonempty interval", "nondegenerate/proper interval". Otherwise one would have to say things like "an interval or a singleton set", "an interval, a singleton, or the empty set" to refer to the broader classes. – Rahul Jan 20 '17 at 20:43

Nothing is lost, although certain theorems/definitions may be more cumbersome to state, and certain nice properties of intervals may no longer hold under this definition. But math wouldn't be seriously affected: nothing prevents us from introducing a new term "psinterval" to refer to what we used to call intervals, and then using that to state/prove things.

The names we assign to concepts have no actual bearing on what we can do with them, or with mathematics (although they may indeed have bearing on how intuitively clear certain things are).

• I agree, but doesn't this point need to be argued somehow? – user159517 Jan 20 '17 at 20:40
• It has been argued: over the years, lots of people doing analysis decided that the current definition of "interval" was useful to them; no one managed to convince anyone that the alternative should be distinguished with a name (or at least not one that I've heard of). – John Hughes Jan 20 '17 at 20:56
• @JohnHughes But note that that's just an argument over which term is better; there's no argument over whether one definition or the other would somehow let us do "more" mathematics. – Noah Schweber Jan 20 '17 at 21:22
• Agreed. I thought your answer addressed that, but I may have misinterpreted user159517's comment. – John Hughes Jan 20 '17 at 21:26

To add to Noah Schweber's answer, the property of convexity is a major nice feature because the family of convex sets in some metric space is closed under arbitrary intersection. That is one reason why it is usually most convenient to allow an interval to be empty. Another reason is that it makes life a lot easier when reasoning about intervals using interval notation defined using the affinely-extended reals denoted by $\def\rr{\mathbb{R}}\def\rrr{\overline{\rr}}$$\rrr$:

$[a,b] = \{ x : x \in \rrr \land a \le x \le b \}$ for any $a,b \in \rrr$.

$[a,b) = \{ x : x \in \rrr \land a \le x < b \}$ for any $a,b \in \rrr$.

$(a,b] = \{ x : x \in \rrr \land a < x \le b \}$ for any $a,b \in \rrr$.

$(a,b) = \{ x : x \in \rrr \land a < x < b \}$ for any $a,b \in \rrr$.

If intervals did not include singletons or the empty-set, then not all of the above notation would refer to intervals. We could of course stipulate that we are not allowed to use the above notation unless $a < b$, but there is good reason not to, because it is convenient to agree with the integer range notation:

$[a{..}b] = [a,b] \cap \mathbb{N}$. Similarly for other intervals.

For integer ranges it is incredibly useful to have $[a{..}a] = \{a\}$ and $[1{..}0] = [a{..}a) = \varnothing$.