Difference between revisions of "Lindelöf Theorem"

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<p>Unfortunately, <math>\mathbb{N}</math> is not countably compact if we use the example open cover which showed that <math>\mathbb{N}</math> was not compact.</p>
 
<p>Unfortunately, <math>\mathbb{N}</math> is not countably compact if we use the example open cover which showed that <math>\mathbb{N}</math> was not compact.</p>
 
<p>So as we can see, the concept of compactness, Lindelöfness, and countable compactness are different properties.</p>
 
<p>So as we can see, the concept of compactness, Lindelöfness, and countable compactness are different properties.</p>
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<h1 id="toc0">The Lindelöf Lemma</h1>
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<p>A topological space <math>X</math> is said to be Lindelöf if every open cover of <math>X</math> has a countable subcover.</p>
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<p>Furthermore, we said that <math>X</math> is countably compact if every countable open cover of <math>X</math> has a finite subcover.</p>
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<p>Awhile back, on the <a href="/second-countable-topological-spaces">Second Countable Topological Spaces</a> page we said that a space <math>X</math> is second countable if there exists a countable basis for the topology on <math>X</math>.</p>
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<p>We will now look at a very important connection between Lindelöf spaces and second countable spaces which we state and prove below.</p>
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<table class="wiki-content-table">
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<tr>
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<td><strong>Lemma 1 (The Lindelöf Lemma):</strong> If <math>X</math> is a second countable topological space then <math>X</math> is Lindelöf.</td>
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</tr>
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</table>
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<ul>
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<li><strong>Proof:</strong> Let <math>X</math> be a second countable topology space. Then there exists a countable basis <math>\mathcal B</math> of the topology <math>\tau</math> on <math>X</math>.</li>
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</ul>
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<ul>
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<li>Now, let <math>\mathcal F</math> be any open cover of <math>X</math> so that:</li>
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</ul>
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<div class="math-equation" id="equation-1">\begin{align} \quad X = \bigcup_{A \in \mathcal F} A \end{align}</div>
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<ul>
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<li>Note that each <math>A \in \mathcal F</math> is an open set since <math>\mathcal F</math> is an open cover of <math>X</math>. Therefore, for each <math>A</math> there exists a subcollection <math>\mathcal B_A \subseteq \mathcal B</math> such that:</li>
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</ul>
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<div class="math-equation" id="equation-2">\begin{align} \quad A = \bigcup_{B \in \mathcal B_A} B \end{align}</div>
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<ul>
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<li>Define an open cover <math>\mathcal C</math> of <math>X</math> as follows:</li>
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</ul>
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<div class="math-equation" id="equation-3">\begin{align} \quad \mathcal C = \left \{ B \in \mathcal B : A = \bigcup_{B \in \mathcal B_A} B, \: \mathrm{for \: some \:} A \in \mathcal F \right \} \end{align}</div>
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<ul>
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<li>Then <math>\mathcal C</math> is countable since <math>\mathcal C \subseteq \mathcal B</math> and <math>\mathcal B</math> is countable.</li>
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</ul>
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<ul>
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<li>We now define an open subcover of <math>\mathcal F</math>. For each element <math>C \in \mathcal C</math>, take an element <math>A \in \mathcal F</math> such that <math>C \subseteq A</math>. This is possible since <math>\mathcal C</math> is a subcollection of the countable basis <math>\mathcal B</math> and every element <math>A \in \mathcal F</math> is the union of some collection of basis elements. Then <math>\mathcal F^*</math> is countable, and moreover:</li>
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</ul>
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<div class="math-equation" id="equation-4">\begin{align} \quad X = \bigcup_{A \in \mathcal F^*} A \end{align}</div>
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<ul>
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<li>Hence every open cover <math>\mathcal F</math> of <math>X</math> has a countable subcover <math>\mathcal F^*</math>. Therefore, <math>X</math> is Lindelöf. <math>\blacksquare</math></li>
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</ul>
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==Licensing==
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Content obtained and/or adapted from:
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* [http://mathonline.wikidot.com/lindeloef-and-countably-compact-topological-spaces Lindelof and Countably Compact Topological Spaces, mathonline.wikidot.com] under a CC BY-SA license
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* [http://mathonline.wikidot.com/the-lindeloef-lemma The Lindelof Lemma, mathonline.wikidot.com] under a CC BY-SA license

Revision as of 13:04, 26 October 2021

Lindelöf and Countably Compact Topological Spaces

If Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X} is a topological space and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A \subseteq X} , then Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A} is said to be compact in if every open cover of has a finite subcover.

Moreover, we said that is a compact topological space if every open cover of has a finite subcover.

We will now look at two similar definitions.

Definition: A topological space is said to be Lindelöf if every open cover of has a countable subcover. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X} is said to be Countably Compact if every countable open cover of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X} has a finite subcover.

It should be noted that the Lindelöf and countably compact property are weaker than the compactness property. If Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X} is a compact topological space then Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X} is also Lindelöf and countably compact.

For example, since Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle [0, 1]} is a compact topological space (with the subspace topology from the usual topology on Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{R}} ), then by extension, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle [0, 1]} is both Lindelöf and countably compact.

Of course, there exists topological spaces which are not compact but are still Lindelöf or countably compact.

For another example, consider the set of natural numbers Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{N}} with the discrete topology, i.e., every subset of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{N}} is open. Then Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{N}} is not compact, because of the following open cover of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{N}} :

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathcal{F} = \{ \{n \} : n \in \mathbb{N} \} = \{ \{ 1 \}, \{ 2 \}, ..., \{ n \}, ... \} }

Clearly there does not exist any subcollection Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathcal F^* \subseteq \mathcal F} that is finite and still covers Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{N}} ! So Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{N}} is not compact.

However, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{N}} is Lindelöf. To show this, let Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathcal F} be any open cover of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{N}} . Then we can choose a countable collection of sets from Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathcal F} which also cover Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{N}} since each subset of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathcal F} has cardinality greater than or equal to Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1} .

Unfortunately, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{N}} is not countably compact if we use the example open cover which showed that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{N}} was not compact.

So as we can see, the concept of compactness, Lindelöfness, and countable compactness are different properties.

The Lindelöf Lemma

A topological space Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X} is said to be Lindelöf if every open cover of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X} has a countable subcover.

Furthermore, we said that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X} is countably compact if every countable open cover of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X} has a finite subcover.

Awhile back, on the <a href="/second-countable-topological-spaces">Second Countable Topological Spaces</a> page we said that a space Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X} is second countable if there exists a countable basis for the topology on Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X} .

We will now look at a very important connection between Lindelöf spaces and second countable spaces which we state and prove below.

Lemma 1 (The Lindelöf Lemma): If Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X} is a second countable topological space then Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X} is Lindelöf.
  • Proof: Let Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X} be a second countable topology space. Then there exists a countable basis Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathcal B} of the topology Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \tau} on Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X} .
  • Now, let Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathcal F} be any open cover of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X} so that:
\begin{align} \quad X = \bigcup_{A \in \mathcal F} A \end{align}
  • Note that each Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A \in \mathcal F} is an open set since Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathcal F} is an open cover of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X} . Therefore, for each there exists a subcollection Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathcal B_A \subseteq \mathcal B} such that:
\begin{align} \quad A = \bigcup_{B \in \mathcal B_A} B \end{align}
  • Define an open cover Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathcal C} of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X} as follows:
\begin{align} \quad \mathcal C = \left \{ B \in \mathcal B : A = \bigcup_{B \in \mathcal B_A} B, \: \mathrm{for \: some \:} A \in \mathcal F \right \} \end{align}
  • Then Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathcal C} is countable since Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathcal C \subseteq \mathcal B} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathcal B} is countable.
  • We now define an open subcover of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathcal F} . For each element Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C \in \mathcal C} , take an element Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A \in \mathcal F} such that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C \subseteq A} . This is possible since Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathcal C} is a subcollection of the countable basis Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathcal B} and every element Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A \in \mathcal F} is the union of some collection of basis elements. Then Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathcal F^*} is countable, and moreover:
\begin{align} \quad X = \bigcup_{A \in \mathcal F^*} A \end{align}
  • Hence every open cover Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathcal F} of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X} has a countable subcover Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathcal F^*} . Therefore, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X} is Lindelöf. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \blacksquare}


Licensing

Content obtained and/or adapted from:

Retrieved from "https://mathresearch.utsa.edu/wiki/index.php?title=Lindelöf_Theorem&oldid=3000"