# Frontiers (Boundaries) in the Plane ...

#### Peter

##### Well-known member
MHB Site Helper
I am reading Stephen Willard: General Topology ... ... and am studying Chapter 2: Topological Spaces and am currently focused on Section 3: Fundamental Concepts ... ...

I need help in order to prove the assertion in Willard Exercise 3B ...

The assertion in Willard Exercise 3B reads as follows:

I am assuming that Willard is assuming the usual topology and metric in $$\displaystyle \mathbb{R}^2$$ ... and so a set $$\displaystyle E$$ is open iff for each $$\displaystyle x \in E$$ there is an $$\displaystyle \epsilon$$-disk (open ball) $$\displaystyle U(x, \epsilon)$$ about x contained in $$\displaystyle E$$ ...

Can someone please rigorously demonstrate the truth of assertion 3B ...

Help will be much appreciated ...

Peter

===============================================================================================================

Readers of the above post may be helped by access to Willard's definition of an $$\displaystyle \epsilon$$-disk (open ball) and an open set in a metric space so I am providing access to the same ... as follows:

Readers of the above post may also be helped by access to Willard's definition of frontier (boundary) and some basic results regarding the frontier of a set .... so I am providing access to the same ... as follows: ...

Readers of the above post may also be helped by access to Willard's definition of closure and some basic results regarding the closure of a set .... so I am providing access to the same ... as follows: ...

Readers of the above post may also be helped by access to Willard's definition of interior and some basic results regarding theinterior of a set .... so I am providing access to the same ... as follows: ...

Hope that helps ...

Peter

#### Attachments

• 60 KB Views: 1
Last edited:

#### Opalg

##### MHB Oldtimer
Staff member
I am reading Stephen Willard: General Topology ... ... and am studying Chapter 2: Topological Spaces and am currently focused on Section 3: Fundamental Concepts ... ...

I need help in order to prove the assertion in Willard Exercise 3B ...

The assertion in Willard Exercise 3B reads as follows:

View attachment 10017

I am assuming that Willard is assuming the usual topology and metric in $$\displaystyle \mathbb{R}^2$$ ... and so a set $$\displaystyle E$$ is open iff for each $$\displaystyle x \in E$$ there is an $$\displaystyle \epsilon$$-disk (open ball) $$\displaystyle U(x, \epsilon)$$ about x contained in $$\displaystyle E$$ ...

Can someone please rigorously demonstrate the truth of assertion 3B ...
To start with a bit of notation, I'll denote the boundary of $E$ (I prefer to call it the boundary rather than the frontier) by $\partial E$.

As far as I can see, this Exercise 3B is not easy. To see why, consider the example where $E$ is the closed unit disk in $\Bbb{R}^2$: $E = \{(x,y) \in \Bbb{R}^2: \|(x,y)\| \leqslant 1\}.$ Its boundary is the unit circle, and its interior is the open unit disc $\{(x,y) \in \Bbb{R}^2: \|(x,y)\| < 1\}.$ Can you think of a set $X\subset \Bbb{R}^2$ such that $E = \partial X$? It took me a while to realise that a possible answer would be to take $X$ to be the set of all elements in $E$ whose coordinates are rational: $X = \{(x,y) \in \Bbb{Q}^2: \|(x,y)\| \leqslant 1\}.$ Then the closure of $X$ is equal to the whole of $E$.

If you can see why that example works, I think it may give a clue to finding the general solution to the problem. If $E$ is a closed set in $\Bbb{R}^2$ then $E$ consists of its boundary together with its interior: $E = E^\circ \cup \partial E$. There is a theorem saying that an open set in $\Bbb{R}^n$ contains a countable dense subset. So let $Y$ be a countable dense subset of $E^\circ$, and let $X = Y \cup \partial E$. Then it should be true that $E = \overline{X}$.

I hope I am not making this problem too complicated. Maybe someone can come up with a simpler approach?

#### Peter

##### Well-known member
MHB Site Helper
Thanks for your analysis Opalg ...

Thanks also for a rather interesting and I must say stunning example ...

I am still puzzling over this exercise...

I note in passing that Willard doesn’t cover countable dense subsets until much later in his book ... so like you I hope someone can come up with a simpler approach ...

Thanks again for the analysis and the example ...

Peter

#### Peter

##### Well-known member
MHB Site Helper
To start with a bit of notation, I'll denote the boundary of $E$ (I prefer to call it the boundary rather than the frontier) by $\partial E$.

As far as I can see, this Exercise 3B is not easy. To see why, consider the example where $E$ is the closed unit disk in $\Bbb{R}^2$: $E = \{(x,y) \in \Bbb{R}^2: \|(x,y)\| \leqslant 1\}.$ Its boundary is the unit circle, and its interior is the open unit disc $\{(x,y) \in \Bbb{R}^2: \|(x,y)\| < 1\}.$ Can you think of a set $X\subset \Bbb{R}^2$ such that $E = \partial X$? It took me a while to realise that a possible answer would be to take $X$ to be the set of all elements in $E$ whose coordinates are rational: $X = \{(x,y) \in \Bbb{Q}^2: \|(x,y)\| \leqslant 1\}.$ Then the closure of $X$ is equal to the whole of $E$.

If you can see why that example works, I think it may give a clue to finding the general solution to the problem. If $E$ is a closed set in $\Bbb{R}^2$ then $E$ consists of its boundary together with its interior: $E = E^\circ \cup \partial E$. There is a theorem saying that an open set in $\Bbb{R}^n$ contains a countable dense subset. So let $Y$ be a countable dense subset of $E^\circ$, and let $X = Y \cup \partial E$. Then it should be true that $E = \overline{X}$.

I hope I am not making this problem too complicated. Maybe someone can come up with a simpler approach?
The more I reflect on your post the more I think you have solved the exercise as simply as is possible...

‘Thanks again

Peter