Countably Dense Subsets in a Metric Space .... Stromberg, Lemma 3.44 .... ....

In summary, a countably dense subset in a metric space is a subset of points that are close together and cover the entire space. It is different from a dense subset in that it must be both dense and countable. Stromberg, Lemma 3.44 states that every countably dense subset can be expanded to a complete metric space, showing their close relationship. To prove that a subset is countably dense, it must be shown to be both dense and countable. A countably dense subset cannot be uncountable, as this would go against its defining characteristics.
  • #1
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I am reading Karl R. Stromberg's book: "An Introduction to Classical Real Analysis". ... ...

I am focused on Chapter 3: Limits and Continuity ... ...

I need help in order to fully understand the proof of Lemma 3.44 on page 105 ... ... Lemma 3.44 and its proof read as follows:

View attachment 9143

In the above proof by Stromberg we read the following:

" ... ... Also, if \(\displaystyle x \in X\) and \(\displaystyle \epsilon \gt 0\), it follows from (2) that \(\displaystyle B_\epsilon (x) \cap A \supset B_{ 1/n } (x) \cap A_{ 1/n } \neq \emptyset\) , where \(\displaystyle 1/n \lt \epsilon\) ... ... "My question is as follows:

Can someone please demonstrate rigorously why/how it is the case that \(\displaystyle B_\epsilon (x) \cap A \supset B_{ 1/n } (x) \cap A_{ 1/n }\) ... ... ?

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*** EDIT ***

After a little reflection this issue may be straightforward ... ... Wish to show formally that \(\displaystyle B_{ 1/n } (x) \cap A_{ 1/n } \subset B_\epsilon (x) \cap A \)We need to show that \(\displaystyle x \in B_{ 1/n } (x) \cap A_{ 1/n } \Longrightarrow x \in B_\epsilon (x) \cap A\)But ... leaving out details ... we have ...\(\displaystyle x \in B_{ 1/n } (x) \cap A_{ 1/n }\)\(\displaystyle \Longrightarrow x \in B_{ 1/n } (x) \text{ and } x \in A_{ 1/n }\) \(\displaystyle \Longrightarrow x \in B_{ \epsilon } (x) \text{ and } x \in A\)\(\displaystyle \Longrightarrow x \in B_\epsilon (x) \cap A\)
Is that correct?

=======================================================================================Help will be appreciated ...

Peter
 

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  • #2
Yes, it is straight forward. In a metric space, where we have a notion of "d(x,y)", the distance between points x and y, a "ball", "$B_\epsilon(x)$" where x is a point of the metric space and $\epsilon$ is a positive real number, is defined as the set of all points, y, in the metric space such that $d(x,y)< \epsilon$. Geometrically it is the interior of the sphere with center at x and radius $\epsilon$.

Geometrically, $B_x(1/n)$ is the ball with center x and radius 1/n while $B_x(\epsilon)$ is the ball with center x and radius $\epsilon$. If $1/n< \epsilon$ then the two balls have the same center but $B_x(1/n)$ has the smaller radius so is completely contained in $B_x(\epsilon)$.

Algebraically, any point, y, ih $B_x(1/n)$ has distance from x, d(x,y)< 1/n. But $1/n< \epsilon$ so $d(x,y)< 1/n< \epsilon$. Since $d(x,y)< \epsilon$, y is also in $B_x(\epsilon)$. Since y could be any point of $B_x(1/n)$, $B_x(1/n)\subset B_x(\epsilon)$.
 
  • #3
By the way, in your title, "Countably Dense Subsets in a Metric Space", "countably" is an adverb modifying "dense". That makes no sense- there is no such thing as "countably dense". You should have "Countable Dense Subsets in a Metric Space" where "countable" is an adjective modifying "subset".
 

Related to Countably Dense Subsets in a Metric Space .... Stromberg, Lemma 3.44 .... ....

1. What is a countably dense subset in a metric space?

A countably dense subset in a metric space is a subset of the metric space that contains a countable number of points and is also dense, meaning that every point in the metric space is either an element of the subset or a limit point of the subset.

2. How is a countably dense subset different from a dense subset?

A countably dense subset is a subset with a countable number of points, while a dense subset can have an uncountable number of points. Additionally, a dense subset must have at least one point in every non-empty open set, while a countably dense subset only needs to have one point in every non-empty open set.

3. What is the significance of Lemma 3.44 in Stromberg's work on countably dense subsets in metric spaces?

Lemma 3.44 in Stromberg's work provides a way to construct a countably dense subset in a metric space by taking a countable union of dense subsets. This is important because it allows us to create a countably dense subset in a metric space that may not have a countable dense subset already.

4. Can a metric space have more than one countably dense subset?

Yes, a metric space can have multiple countably dense subsets. For example, in the metric space of real numbers, the set of rational numbers and the set of irrational numbers are both countably dense subsets.

5. How are countably dense subsets used in mathematical analysis?

Countably dense subsets are used in mathematical analysis to approximate uncountable sets. By constructing a countably dense subset, we can make statements and prove theorems about the entire uncountable set. They are also useful in proving the existence of certain types of functions, such as continuous functions, on metric spaces.

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