Lindelöf Theorem

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Lindelöf and Countably Compact Topological Spaces

If 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 is a compact topological space (with the subspace topology from the usual topology on ), then by extension, 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 with the discrete topology, i.e., every subset of is open. Then is not compact, because of the following open cover of :

Clearly there does not exist any subcollection that is finite and still covers ! So is not compact.

However, is Lindelöf. To show this, let be any open cover of . Then we can choose a countable collection of sets from which also cover since each subset of has cardinality greater than or equal to .

Unfortunately, is not countably compact if we use the example open cover which showed that 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 is said to be Lindelöf if every open cover of has a countable subcover.

Furthermore, we said that is countably compact if every countable open cover of has a finite subcover.

A space is second countable if there exists a countable basis for the topology on .

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 is a second countable topological space then is Lindelöf.

  • Proof: Let be a second countable topology space. Then there exists a countable basis of the topology on .
  • Now, let be any open cover of so that:

  • Note that each is an open set since is an open cover of . Therefore, for each there exists a subcollection such that:

  • Define an open cover of as follows:

  • Then is countable since and is countable.
  • We now define an open subcover of . For each element , take an element such that . This is possible since is a subcollection of the countable basis and every element is the union of some collection of basis elements. Then is countable, and moreover:

  • Hence every open cover of has a countable subcover . Therefore, is Lindelöf.

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