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,

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