Difference between revisions of "Lindelöf Theorem"
| Line 12: | Line 12: | ||
<p>Of course, there exists topological spaces which are not compact but are still Lindelöf or countably compact.</p> | <p>Of course, there exists topological spaces which are not compact but are still Lindelöf or countably compact.</p> | ||
<p>For another example, consider the set of natural numbers <math>\mathbb{N}</math> with the discrete topology, i.e., every subset of <math>\mathbb{N}</math> is open. Then <math>\mathbb{N}</math> is not compact, because of the following open cover of <math>\mathbb{N}</math>:</p> | <p>For another example, consider the set of natural numbers <math>\mathbb{N}</math> with the discrete topology, i.e., every subset of <math>\mathbb{N}</math> is open. Then <math>\mathbb{N}</math> is not compact, because of the following open cover of <math>\mathbb{N}</math>:</p> | ||
| − | + | ||
| − | <math>\begin{ | + | <math>\begin{align} \mathcal F = \{ \{n \} : n \in \mathbb{N} \} = \{ \{ 1 \}, \{ 2 \}, ..., \{ n \}, ... \} \end{align}</math> |
| + | |||
<p>Clearly there does not exist any subcollection <math>\mathcal F^* \subseteq \mathcal F</math> that is finite and still covers <math>\mathbb{N}</math>! So <math>\mathbb{N}</math> is not compact.</p> | <p>Clearly there does not exist any subcollection <math>\mathcal F^* \subseteq \mathcal F</math> that is finite and still covers <math>\mathbb{N}</math>! So <math>\mathbb{N}</math> is not compact.</p> | ||
<p>However, <math>\mathbb{N}</math> is Lindelöf. To show this, let <math>\mathcal F</math> be any open cover of <math>\mathbb{N}</math>. Then we can choose a countable collection of sets from <math>\mathcal F</math> which also cover <math>\mathbb{N}</math> since each subset of <math>\mathcal F</math> has cardinality greater than or equal to <math>1</math>.</p> | <p>However, <math>\mathbb{N}</math> is Lindelöf. To show this, let <math>\mathcal F</math> be any open cover of <math>\mathbb{N}</math>. Then we can choose a countable collection of sets from <math>\mathcal F</math> which also cover <math>\mathbb{N}</math> since each subset of <math>\mathcal F</math> has cardinality greater than or equal to <math>1</math>.</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>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> | ||
Revision as of 12:55, 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 , then 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. is said to be Countably Compact if every countable open cover of has a finite subcover.
It should be noted that the Lindelöf and countably compact property are weaker than the compactness property. If is a compact topological space then 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 \begin{align} \mathcal F = \{ \{n \} : n \in \mathbb{N} \} = \{ \{ 1 \}, \{ 2 \}, ..., \{ n \}, ... \} \end{align}}
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.
