Separable Metric Spaces - Revision history

Lila at 17:54, 8 November 2021

2021-11-08T17:54:21Z

Lila: Created page with "===Dense Sets in a Metric Space===
We will now look at a new concept regarding metric spaces known as dense sets which we define below.

2021-11-08T17:53:03Z

Created page with "===Dense Sets in a Metric Space=== We will now look at a new concept regarding metric spaces known as dense sets which we define below. <blockquote style="background: w..."

New page
===Dense Sets in a Metric Space===
We will now look at a new concept regarding metric spaces known as dense sets which we define below.
<blockquote style="background: white; border: 1px solid black; padding: 1em;">
<td>Definition: Let <math>(M, d)</math> be a metric space and let <math>S \subseteq M</math>. Then <math>S</math> is said to be Dense in <math>M</math> if for every <math>x \in M</math> and for every <math>r > 0</math> we have that <math>B(x, r) \cap S \neq \emptyset</math>, i.e., every open ball in <math>(M, d)</math> contains a point of <math>S</math>.</td>
</blockquote>
In any metric space <math>(M, d)</math> the whole set <math>M</math> is always dense in <math>M</math>. Furthermore, the empty set <math>\emptyset</math> is not dense in <math>M</math>.
For a less trivial example, consider the metric space <math>(\mathbb{R}, d)</math> where <math>d</math> is the usual Euclidean metric defined for all <math>x, y \in \mathbb{R}</math> by <math>d(x, y) = \mid x - y \mid</math>, and consider the subset <math>\mathbb{Q} \subset \mathbb{R}</math> of rational numbers.
The set <math>\mathbb{Q}</math> is dense in <math>\mathbb{R}</math> because for any open ball, i.e., for any <math>x \in \mathbb{R}</math> and for any <math>r > 0</math> we have that the open interval <math>(x - r, x + r)</math> contains a rational number as we saw on <a class="newpage" href="/the-density-of-real-numbers-theorem">The Density of Real Numbers Theorem</a> page.
For a counterexample, consider the set <math>\mathbb{Z} \subset \mathbb{R}</math> of integers. We claim that <math>\mathbb{Z}</math> is not dense in <math>\mathbb{R}</math>. To show this, consider the following ball:
<div style="text-align: center;"><math>\begin{align} \quad B \left ( \frac{1}{2}, \frac{1}{2} \right ) = (0, 1) \end{align}</math></div>
Clearly <math>\mathbb{Z} \cap (0, 1) = \emptyset</math> and so <math>\mathbb{Z}</math> is not dense in <math>\mathbb{R}</math>.
We will now look at a nice theorem which tells us that for a metric space <math>(M, d)</math> a set <math>S \subseteq M</math> is dense in <math>M</math> if and only if its closure equals <math>M</math>.
<blockquote style="background: white; border: 1px solid black; padding: 1em;">
<td>Theorem 1: Let <math>(M, d)</math> be a metric space and let <math>S \subseteq M</math>. Then, <math>S</math> is dense in <math>M</math> if and only if <math>\bar{S} = M</math>.</td>
</blockquote>
Recall from the <a href="/adherent-accumulation-and-isolated-points-in-metric-spaces">Adherent, Accumulation, and Isolated Points in Metric Spaces</a> page that <math>\bar{S}</math> denotes the closure of <math>S</math>, and we defined the closure of <math>S</math> to be the set of adherent points of <math>S</math>.
<ul>
<li>Proof: <math>\Rightarrow</math> Suppose that <math>S</math> is dense in <math>M</math>. Then for all <math>x \in M</math> and all <math>r > 0</math> we have that:</li>
</ul>
<div style="text-align: center;"><math>\begin{align} \quad B(x, r) \cap S \neq \emptyset \end{align}</math></div>
<ul>
<li>So every <math>x \in M</math> is an adherent point of <math>S</math>. The set of all adherent points of <math>S</math> is the closure of <math>S</math>, so <math>\bar{S} = M</math>.</li>
</ul>
<ul>
<li><math>\Leftarrow</math> Suppose that <math>\bar{S} = M</math>. Then every point of <math>M</math> is an adherent point of <math>S</math>, i.e., for all <math>x \in M</math> and for all <math>r > 0</math> we have that:</li>
</ul>
<div style="text-align: center;"><math>\begin{align} \quad B(x, r) \cap S \neq \emptyset \end{align}</math></div>
<ul>
<li>Therefore <math>S</math> is dense in <math>M</math>. <math>\blacksquare</math></li>
</ul>

===Separable Metric Spaces===
Recall from the <a href="/dense-sets-in-a-metric-space">Dense Sets in a Metric Space</a> page that if <math>(M, d)</math> is a metric space then a subset <math>S \subseteq M</math> is said to be dense in <math>M</math> if for every <math>x \in M</math> and for all <math>r > 0</math> we have that:
<div style="text-align: center;"><math>\begin{align} \quad B(x, r) \cap S \neq \emptyset \end{align}</math></div>
In other words, <math>S</math> is dense in <math>M</math> if every open ball contains a point of <math>S</math>.
We will now look at a special type of metric space known as a separable metric space which we define below.
<blockquote style="background: white; border: 1px solid black; padding: 1em;">
<td>Definition: A metric space <math>(M, d)</math> is said to be Separable if there exists a countable dense subset <math>S</math> of <math>M</math>.</td>
</blockquote>
For example, consider the metric space <math>(\mathbb{R}, d)</math> where <math>d</math> is the usual Euclidean metric defined for all <math>x, y \in \mathbb{R}</math> by <math>d(x, y) = \mid x - y \mid</math>. Then the subset <math>\mathbb{Q}</math> is dense in <math>\mathbb{R}</math> since every open interval contains rational numbers.
In fact, in general, the metric space <math>(\mathbb{R}^n, d)</math> where <math>d</math> is the usual Euclidean metric defined for all <math>\mathbf{x} = (x_1, x_2, ..., x_n), \mathbf{y} = (y_1, y_2, ..., y_n) \in \mathbb{R}^n</math> by:
<div style="text-align: center;"><math>\begin{align} \quad \| \mathbf{x} - \mathbf{y} \| = \sqrt{(x_1 - y_1)^2 + (x_2 - y_2)^2 + ... + (x_n - y_n)^2} \end{align}</math></div>
Then it can be shown similarly that the following set is dense in <math>\mathbb{R}^n</math>:
<div style="text-align: center;"><math>\begin{align} \quad \mathbb{Q}^n = \{(x_1, x_2, ..., x_n) \in \mathbb{R}^n : x_1, x_2, ..., x_n \in \mathbb{Q} \} \end{align}</math></div>

== Licensing ==
Content obtained and/or adapted from:
* [http://mathonline.wikidot.com/dense-sets-in-a-metric-space Dense Sets in a Metric Space, mathonline.wikidot.com] under a CC BY-SA license
* [http://mathonline.wikidot.com/separable-metric-spaces Separable Metric Spaces, mathonline.wikidot.com] under a CC BY-SA license

@@ Line 6: / Line 6: @@
 <p>In any metric space <span class="math-inline"><math>(M, d)</math></span> the whole set <span class="math-inline"><math>M</math></span> is always dense in <span class="math-inline"><math>M</math></span>. Furthermore, the empty set <span class="math-inline"><math>\emptyset</math></span> is not dense in <span class="math-inline"><math>M</math></span>.</p>
 <p>For a less trivial example, consider the metric space <span class="math-inline"><math>(\mathbb{R}, d)</math></span> where <span class="math-inline"><math>d</math></span> is the usual Euclidean metric defined for all <span class="math-inline"><math>x, y \in \mathbb{R}</math></span> by <span class="math-inline"><math>d(x, y) = \mid x - y \mid</math></span>, and consider the subset <span class="math-inline"><math>\mathbb{Q} \subset \mathbb{R}</math></span> of rational numbers.</p>
-<p>The set <span class="math-inline"><math>\mathbb{Q}</math></span> is dense in <span class="math-inline"><math>\mathbb{R}</math></span> because for any open ball, i.e., for any <span class="math-inline"><math>x \in \mathbb{R}</math></span> and for any <span class="math-inline"><math>r > 0</math></span> we have that the open interval <span class="math-inline"><math>(x - r, x + r)</math></span> contains a rational number as we saw on <a class="newpage" href="/the-density-of-real-numbers-theorem">The Density of Real Numbers Theorem</a> page.</p>
+<p>The set <span class="math-inline"><math>\mathbb{Q}</math></span> is dense in <span class="math-inline"><math>\mathbb{R}</math></span> because for any open ball, i.e., for any <span class="math-inline"><math>x \in \mathbb{R}</math></span> and for any <span class="math-inline"><math>r > 0</math></span> we have that the open interval <span class="math-inline"><math>(x - r, x + r)</math></span> contains a rational number.</p>
 <p>For a counterexample, consider the set <span class="math-inline"><math>\mathbb{Z} \subset \mathbb{R}</math></span> of integers. We claim that <span class="math-inline"><math>\mathbb{Z}</math></span> is not dense in <span class="math-inline"><math>\mathbb{R}</math></span>. To show this, consider the following ball:</p>
 <div style="text-align: center;"><math>\begin{align} \quad B \left ( \frac{1}{2}, \frac{1}{2} \right ) = (0, 1) \end{align}</math></div>
@@ Line 14: / Line 14: @@
 <td><strong>Theorem 1:</strong> Let <span class="math-inline"><math>(M, d)</math></span> be a metric space and let <span class="math-inline"><math>S \subseteq M</math></span>. Then, <span class="math-inline"><math>S</math></span> is dense in <span class="math-inline"><math>M</math></span> if and only if <span class="math-inline"><math>\bar{S} = M</math></span>.</td>
 </blockquote>
-<p><em>Recall from the <a href="/adherent-accumulation-and-isolated-points-in-metric-spaces">Adherent, Accumulation, and Isolated Points in Metric Spaces</a> page that <span class="math-inline"><math>\bar{S}</math></span> denotes the closure of <span class="math-inline"><math>S</math></span>, and we defined the closure of <span class="math-inline"><math>S</math></span> to be the set of adherent points of <span class="math-inline"><math>S</math></span>.</em></p>
+<p><em>Recall that <span class="math-inline"><math>\bar{S}</math></span> denotes the closure of <span class="math-inline"><math>S</math></span>, and we defined the closure of <span class="math-inline"><math>S</math></span> to be the set of adherent points of <span class="math-inline"><math>S</math></span>.</em></p>
 <ul>
 <li><strong>Proof:</strong> <span class="math-inline"><math>\Rightarrow</math></span> Suppose that <span class="math-inline"><math>S</math></span> is dense in <span class="math-inline"><math>M</math></span>. Then for all <span class="math-inline"><math>x \in M</math></span> and all <span class="math-inline"><math>r > 0</math></span> we have that:</li>
@@ Line 31: / Line 31: @@
 ===Separable Metric Spaces===
-<p>Recall from the <a href="/dense-sets-in-a-metric-space">Dense Sets in a Metric Space</a> page that if <span class="math-inline"><math>(M, d)</math></span> is a metric space then a subset <span class="math-inline"><math>S \subseteq M</math></span> is said to be dense in <span class="math-inline"><math>M</math></span> if for every <span class="math-inline"><math>x \in M</math></span> and for all <span class="math-inline"><math>r > 0</math></span> we have that:</p>
+<p>Recall that if <span class="math-inline"><math>(M, d)</math></span> is a metric space then a subset <span class="math-inline"><math>S \subseteq M</math></span> is said to be dense in <span class="math-inline"><math>M</math></span> if for every <span class="math-inline"><math>x \in M</math></span> and for all <span class="math-inline"><math>r > 0</math></span> we have that:</p>
 <div style="text-align: center;"><math>\begin{align} \quad B(x, r) \cap S \neq \emptyset \end{align}</math></div>
 <p>In other words, <span class="math-inline"><math>S</math></span> is dense in <span class="math-inline"><math>M</math></span> if every open ball contains a point of <span class="math-inline"><math>S</math></span>.</p>