# open subsets

#### dwsmith

##### Well-known member
If $S\subseteq\mathbb{R}^n$, prove that $S$ is the union of all open subsets $\mathbb{R}^n$ which are contained in $S$.
This is described by saying that int $S$ is the largest open subset of $S$.

I don't really get this.

#### Plato

##### Well-known member
MHB Math Helper
If $S\subseteq\mathbb{R}^n$, prove that $S$ is the union of all open subsets $\mathbb{R}^n$ which are contained in $S$.
This is described by saying that int $S$ is the largest open subset of $S$.
I don't really get this.
The way you have written the statement makes it false.
Do you mean that If $S$ is open then... ?

#### dwsmith

##### Well-known member
The way you have written the statement makes it false.
Do you mean that If $S$ is open then... ?
That is exactly how it is written in Apostles. Problem 3.10

#### Plato

##### Well-known member
MHB Math Helper
That is exactly how it is written in Apostles. Problem 3.10
As written is is still false.
If $S$ has a boundary point $S$ cannot be the union of open sets.
The statement that $S\subseteq \mathbb{R}^n$ means any set in $\mathbb{R}^n$.

The statement is true for any open set. For you see that an open set is its own interior.
That is, every open sets is the union of open balls.

#### dwsmith

##### Well-known member
If $S\subseteq\mathbb{R}^n$, prove that the int $S$ is the union of all open subsets $\mathbb{R}^n$ which are contained in $S$.
This is described by saying that int $S$ is the largest open subset of $S$.

I don't really get this.
I just re-read it and I forgot the int.

#### Plato

##### Well-known member
MHB Math Helper
I just re-read it and I forgot the int.
If $t\in\text{Int}(S)$ then there is an open set $O_t$ such that $t\in O_t~\&~O_t\subseteq S$.
Now finish it.

#### dwsmith

##### Well-known member
If $t\in\text{Int}(S)$ then there is an open set $O_t$ such that $t\in O_t~\&~O_t\subseteq S$.
Now finish it.
I understand what you have written. How can int $S$ be the union of all open sets of $\mathbb{R}^n$. But $\mathbb{R}^n$ can be bigger than $S$.

#### Ackbach

##### Indicium Physicus
Staff member
I understand what you have written. How can int $S$ be the union of all open sets of $\mathbb{R}^n$. But $\mathbb{R}^n$ can be bigger than $S$.
Where did Plato say that?

By the way, I'm guessing that the problem statement is as follows:

If $S\subseteq \mathbb{R}^{n}$, prove that $\text{int}(S)$ is the union of all open subsets of $\mathbb{R}^{n}$ that are contained in $S$.

Is that correct? If so, then it's not that $\text{int}(S)$ is the union of all open subsets of $\mathbb{R}^{n}$. It's the union of all open subsets of $\mathbb{R}^{n}$ that are also subsets of $S$. That is,

$$\text{int}(S)=\bigcup_{x}\:\{x|x\subseteq S\land x\;\text{is open}\}.$$