Open Sets in R .... .... Willard, Example 2.7 (a) .... ....

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In summary, the text discusses the topic of equivalence relations on open sets in $\mathbb{R}$ and how they result in disjoint open intervals whose union is the open set. The text also explains the reasoning behind why there can only be countably many such intervals.
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I am reading Stephen Willard: General Topology ... ... and am currently focused on Chapter 1: Set Theory and Metric Spaces and am currently focused on Section 2: Metric Spaces ... ...

I need help in order to fully understand Example 2.7(a) ... .. The relevant text reads as follows:View attachment 9688My questions are as follows:Question 1

In the above example from Willard we read the following;

" ... ... If \(\displaystyle A\) is an open set in \(\displaystyle \mathbb{R}\), the relation \(\displaystyle x \sim y\) iff there is some open interval \(\displaystyle (a, b)\) with \(\displaystyle \{ x, y \} \subset (a, b) \subset A\) is an equivalence relation on \(\displaystyle A\) and the resulting equivalence classes are disjoint open intervals whose union is \(\displaystyle A\) ... ... "Can someone please demonstrate formally and rigorously that the resulting equivalence classes are disjoint open intervals whose union is \(\displaystyle A\) ... ... ?Question 2

In the above example from Willard we read the following;

" ... ... The fact that there can be only countably many follows since each must contain a distinct rational ... "

I am somewhat lost in trying to understand this statement ... can someone please explain the meaning of "there can be only countably many follows since each must contain a distinct rational ...?
Help will be much appreciated ... ...

Peter
 

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  • #2
Hi Peter,

This is a useful result, so it's good to go through it in some detail.

Question 1

Fix one of the equivalence classes and let this set be represented by $p$; i.e., consider $[p].$

To see that $[p]$ has the interval property, let $x,y\in [p]$, with $x<y$. Since $x\sim y$, there is an open interval $(a,b)$ such that $\{x,y\}\subset (a,b)\subset A$. Since $(a,b)$ is an interval, it follows that $(x,y)\subset (a,b)$. Hence, for any $t$ such that $x<t<y$, $\{x,t\}\subset (a,b)\subset A.$ Hence $x\sim t$, and, therefore $p\sim t$ via transitivity of $\sim$. This proves that $[p]$ has the interval property. (Note: we could have also argued that $t\sim y$ to get the same result).

To prove that $[p]$ is open, let $x\in [p]$. Since $x\in A$ and $A$ is open, there is an open interval $(a,b)$ such that $x\in (a,b)\subset A$. It follows that $x\sim y$ for all $y\in (a,b)$. By transitivity, we have $p\sim y$ for all $y\in (a,b)$, from which we conclude that $x\in (a,b)\subset [p]$. Hence, $[p]$ is open.

Question 2

Because the rationals are dense in $\mathbb{R}$, each of the disjoint open intervals from above contains a rational number (in fact, each of the intervals contains infinitely many rationals but that's not important here). From each interval select a single rational number. Since the intervals are disjoint, it follows that the collection of disjoint open intervals is in bijection with a subset of $\mathbb{Q}.$ Since $\mathbb{Q}$ is countable, there can only be countably many such intervals.Hopefully that helps. Feel free to let me know if anything is still unclear.
 
  • #3
GJA said:
Hi Peter,

This is a useful result, so it's good to go through it in some detail.

Question 1

Fix one of the equivalence classes and let this set be represented by $p$; i.e., consider $[p].$

To see that $[p]$ has the interval property, let $x,y\in [p]$, with $x<y$. Since $x\sim y$, there is an open interval $(a,b)$ such that $\{x,y\}\subset (a,b)\subset A$. Since $(a,b)$ is an interval, it follows that $(x,y)\subset (a,b)$. Hence, for any $t$ such that $x<t<y$, $\{x,t\}\subset (a,b)\subset A.$ Hence $x\sim t$, and, therefore $p\sim t$ via transitivity of $\sim$. This proves that $[p]$ has the interval property. (Note: we could have also argued that $t\sim y$ to get the same result).

To prove that $[p]$ is open, let $x\in [p]$. Since $x\in A$ and $A$ is open, there is an open interval $(a,b)$ such that $x\in (a,b)\subset A$. It follows that $x\sim y$ for all $y\in (a,b)$. By transitivity, we have $p\sim y$ for all $y\in (a,b)$, from which we conclude that $x\in (a,b)\subset [p]$. Hence, $[p]$ is open.

Question 2

Because the rationals are dense in $\mathbb{R}$, each of the disjoint open intervals from above contains a rational number (in fact, each of the intervals contains infinitely many rationals but that's not important here). From each interval select a single rational number. Since the intervals are disjoint, it follows that the collection of disjoint open intervals is in bijection with a subset of $\mathbb{Q}.$ Since $\mathbb{Q}$ is countable, there can only be countably many such intervals.Hopefully that helps. Feel free to let me know if anything is still unclear.
Thanks so much for the help GJA ...

You write:

" ... ... This proves that $[p]$ has the interval property. ... ..."

I have a followup question: How do we show that the equivalence class are disjoint intervals whose union is A?PeterEDIT! Just realized! Equivalence classes are disjoint! and together make up the whole set!

Is that correct?

Peter
 
  • #4
Peter said:
EDIT! Just realized! Equivalence classes are disjoint! and together make up the whole set!

Is that correct?

Peter

That's correct, nicely done!
 

Related to Open Sets in R .... .... Willard, Example 2.7 (a) .... ....

1. What are open sets in R?

Open sets in R are subsets of the real numbers that do not include their boundary points. In other words, for any point in an open set, there exists a small "ball" around that point that is entirely contained within the set.

2. How are open sets different from closed sets in R?

Closed sets in R include their boundary points, while open sets do not. This means that for a closed set, every point on the boundary is also included in the set, while for an open set, the boundary points are not included.

3. What is the significance of open sets in R?

Open sets are important in the study of topology and analysis in mathematics. They allow for the definition of limit points, continuity, and other important concepts. They also help to define the topology of a space, which is important in understanding the properties of that space.

4. Can you give an example of an open set in R?

One example of an open set in R is the interval (0,1), which contains all real numbers between 0 and 1, but does not include the endpoints 0 and 1. Another example is the set of all positive real numbers, (0,∞).

5. How are open sets defined in Willard's Example 2.7 (a)?

In Willard's Example 2.7 (a), open sets are defined as sets that do not contain any of their boundary points. This definition is consistent with the general definition of open sets in topology and analysis.

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