# Prove: the empty set $\emptyset$ and $\mathbb{R}$ are closed. [closed]

Prove: the empty set $$\emptyset$$ and $$\mathbb{R}$$ are closed.

Definition: Let $$S \subset \mathbb{R}$$.

1. S is said to be open if every point of S is an interior point of S.

2. S is said to be closed if and only if $$\mathbb{R} \setminus S$$ is open.

Proof: $$\mathbb{R}$$ is closed since $$\mathbb{R} \setminus \mathbb{R}$$ = $$\emptyset$$ is open

$$\emptyset$$ is closed since $$\mathbb{R}$$ \ $$\emptyset$$=$$\mathbb{R}$$ is open.

Is it right?

• While $\emptyset$ and $\mathbb{R}$ are closed, they are also open. Mar 19, 2020 at 13:30

$$\mathbb{R}$$ is closed since $$\mathbb{R} \setminus \mathbb{R}$$ = $$\emptyset$$ is open
While it is true that $$\emptyset$$ is trivially clopen (both open and closed), it would help to understand why by applying the definition
This means that: $$\forall \ x \ \in S, \ \exists r \in \mathbb{R^+}, \ b(x;r) \subset S$$, where $$b(x;r)$$ is the open ball of radius $$r$$ centered at $$x$$.
Since $$\emptyset$$ has no $$x$$, the above statement is trivially true and thus $$\emptyset$$ is open.