# Rewriting empty set

#### burt

##### Junior Member
Technically, does this notation (3,3) mean the same thing as empty set?

#### Romsek

##### Full Member
My first impression would be no, why would it. But maybe you can provide further context.

Oh I guess you mean an open interval... interesting... yes I guess that would be equivalent to the empty set.

#### burt

##### Junior Member
provide further context
Doesn't that mean the interval from 3 to 3, not including 3?

#### pka

##### Elite Member
Doesn't that mean the interval from 3 to 3, not including 3?
The statement that $$\displaystyle (a,b)$$ is an open interval in the set of real numbers means that $$\displaystyle a~\&~b$$ are two real numbers and $$\displaystyle a<b$$.
$$\displaystyle (a,b)=\{x\in\mathbb{R}: a<x<b\}$$

#### Romsek

##### Full Member
The statement that $$\displaystyle (a,b)$$ is an open interval in the set of real numbers means that $$\displaystyle a~\&~b$$ are two real numbers and $$\displaystyle a<b$$.
$$\displaystyle (a,b)=\{x\in\mathbb{R}: a<x<b\}$$
and thus $$\displaystyle (a,a) = \{x \in \mathbb{R} : a < x < a\} = \emptyset$$

#### pka

##### Elite Member
and thus $$\displaystyle (a,a) = \{x \in \mathbb{R} : a < x < a\} = \emptyset$$
Anyone who can count knows that $$\displaystyle a=a$$ so $$\displaystyle (a,a)$$ does have two real numbers therefore by definition is not notation for an open interval.

#### burt

##### Junior Member
Anyone who can count knows that $$\displaystyle a=a$$ so $$\displaystyle (a,a)$$ does have two real numbers therefore by definition is not notation for an open interval.
But [a,a] is.

#### Romsek

##### Full Member
But [a,a] is.
no, that's a closed interval. It contains it's single limit point a