The preceding post is an exercise showing that the product of countably many -compact spaces is a Lindelof space. The result is an example of a situation where the Lindelof property is countably productive if each factor is a “nice” Lindelof space. In this case, “nice” means
-compact. This post gives several exercises surrounding the notion of
-compactness.
Exercise 2.A
According to the preceding exercise, the product of countably many -compact spaces is a Lindelof space. Give an example showing that the result cannot be extended to the product of uncountably many
-compact spaces. More specifically, give an example of a product of uncountably many
-compact spaces such that the product space is not Lindelof.
Exercise 2.B
Any -compact space is Lindelof. Since
, the real line with the usual Euclidean topology is
-compact. This exercise is to find an example of “Lindelof does not imply
-compact.” Find one such example among the subspaces of the real line. Note that as a subspace of the real line, the example would be a separable metric space, hence would be a Lindelof space.
Exercise 2.C
This exercise is also to look for an example of a space that is Lindelof and not -compact. The example sought is a non-metric one, preferably a space whose underlying set is the real line and whose topology is finer than the Euclidean topology.
Exercise 2.D
Show that the product of two Lindelof spaces is a Lindelof space whenever one of the factors is a -compact space.
Exercise 2.E
Prove that the product of finitely many -compact spaces is a
-compact space. Give an example of a space showing that the product of countably and infinitely many
-compact spaces does not have to be
-compact. For example, show that
, the product of countably many copies of the real line, is not
-compact.
Comments
The Lindelof property and -compactness are basic topological notions. The above exercises are natural questions based on these two basic notions. One immediate purpose of these exercises is that they provide further interaction with the two basic notions. More importantly, working on these exercise give exposure to mathematics that is seemingly unrelated to the two basic notions. For example, finding
-compactness on subspaces of the real line and subspaces of compact spaces naturally uses a Baire category argument, which is a deep and rich topic that finds uses in multiple areas of mathematics. For this reason, these exercises present excellent learning opportunities not only in topology but also in other useful mathematical topics.
If preferred, the exercises can be attacked head on. The exercises are also intended to be a guided tour. Hints are also provided below. Two sets of hints are given – Hints (blue dividers) and Further Hints (maroon dividers). The proofs of certain key facts are also given (orange dividers). Concluding remarks are given at the end of the post.
Hints for Exercise 2.A
Prove that the Lindelof property is hereditary with respect to closed subspaces. That is, if is a Lindelof space, then every closed subspace of
is also Lindelof.
Prove that if is a Lindelof space, then every closed and discrete subset of
is countable (every space that has this property is said to have countable extent).
Show that the product of uncountably many copies of the real line does not have countable extent. Specifically, focus on either one of the following two examples.
- Show that the product space
has a closed and discrete subspace of cardinality continuum where
is cardinality of continuum. Hence
is not Lindelof.
- Show that the product space
has a closed and discrete subspace of cardinality
where
is the first uncountable ordinal. Hence
is not Lindelof.
Hints for Exercise 2.B
Let be the set of all irrational numbers. Show that
as a subspace of the real line is not
-compact.
Hints for Exercise 2.C
Let be the real line with the topology generated by the half open and half closed intervals of the form
. The real line with this topology is called the Sorgenfrey line. Show that
is Lindelof and is not
-compact.
Hints for Exercise 2.D
It is helpful to first prove: the product of two Lindelof space is Lindelof if one of the factors is a compact space. The Tube lemma is helpful.
Tube Lemma
Let be a space. Let
be a compact space. Suppose that
is an open subset of
and suppose that
where
. Then there exists an open subset
of
such that
.
Hints for Exercise 2.E
Since the real line is homeomorphic to the open interval
,
is homeomorphic to
. Show that
is not
-compact.
Further Hints for Exercise 2.A
The hints here focus on the example .
Let . Let
be the first infinite ordinal. For convenience, consider
the set
, the set of all non-negative integers. Since
is a closed subset of
, any closed and discrete subset of
is a closed and discrete subset of
. The task at hand is to find a closed and discrete subset of
. To this end, we define
after setting up background information.
For each , choose a sequence
of open intervals (in the usual topology of
) such that
,
for each
(the closure is in the usual topology of
).
Note. For each , the open intervals
are of the form
. For
, the open intervals
are of the form
. For
, the open intervals
are of the form
.
For each , define the map
as follows:
We are now ready to define . For each
,
is the mapping
defined by
for each
.
Show the following:
- The set
has cardinality continuum.
- The set
is a discrete space.
- The set
is a closed subspace of
.
Further Hints for Exercise 2.B
A subset of the real line
is nowhere dense in
if for any nonempty open subset
of
, there is a nonempty open subset
of
such that
. If we replace open sets by open intervals, we have the same notion.
Show that the real line with the usual Euclidean topology cannot be the union of countably many closed and nowhere dense sets.
Further Hints for Exercise 2.C
Prove that if and
are
-compact, then the product
is
-compact, hence Lindelof.
Prove that , the Sorgenfrey line, is Lindelof while its square
is not Lindelof.
Further Hints for Exercise 2.D
As suggested in the hints given earlier, prove that is Lindelof if
is Lindelof and
is compact. As suggested, the Tube lemma is a useful tool.
Further Hints for Exercise 2.E
The product space is a subspace of the product space
. Since
is compact, we can fall back on a Baire category theorem argument to show why
cannot be
-compact. To this end, we consider the notion of Baire space. A space
is said to be a Baire space if for each countable family
of open and dense subsets of
, the intersection
is a dense subset of
. Prove the following results.
Fact E.1
Let be a compact Hausdorff space. Let
be a sequence of non-empty open subsets of
such that
for each
. Then the intersection
is non-empty.
Fact E.2
Any compact Hausdorff space is Baire space.
Fact E.3
Let be a Baire space. Let
be a dense
-subset of
such that
is a dense subset of
. Then
is not a
-compact space.
Since is compact, it follows from Fact E.2 that the product space
is a Baire space.
Fact E.4
Let and
. The product space
is a dense
-subset of
. Furthermore,
is a dense subset of
.
It follows from the above facts that the product space cannot be a
-compact space.
Proofs of Key Steps for Exercise 2.A
The proof here focuses on the example .
To see that has the same cardinality as that of
, show that
for
. This follows from the definition of the mapping
.
To see that is discrete, for each
, consider the open set
. Note that
. Further note that
for all
.
To see that is a closed subset of
, let
such that
. Consider two cases.
Case 1. for all
.
Note that is an open cover of
(in the usual topology). There exists a finite
such that
is a cover of
. Consider the open set
. Define the set
as follows:
The set can be further described as follows:
The last step is because
is a cover of
. The fact that
means that
is an open subset of
containing the point
such that
contains no point of
.
Case 2. for some
.
Since ,
for all
. In particular,
. This means that
for some
. Define the open set
as follows:
Clearly . Observe that
since
. For each
,
since
. Thus
is an open set containing
such that
.
Both cases show that is a closed subset of
.
Proofs of Key Steps for Exercise 2.B
Suppose that , the set of all irrational numbers, is
-compact. That is,
where each
is a compact space as a subspace of
. Any compact subspace of
is also a compact subspace of
. As a result, each
is a closed subset of
. Furthermore, prove the following:
-
Each
Each singleton set where
is any rational number is also a closed and nowhere dense subset of
. This means that the real line is the union of countably many closed and nowhere dense subsets, contracting the hints given earlier. Thus
cannot be
-compact.
Proofs of Key Steps for Exercise 2.C
The Sorgenfrey line is a Lindelof space whose square
is not normal. This is a famous example of a Lindelof space whose square is not Lindelof (not even normal). For reference, a proof is found here. An alternative proof of the non-normality of
uses the Baire category theorem and is found here.
If the Sorgenfrey line is -compact, then
would be
-compact and hence Lindelof. Thus
cannot be
-compact.
Proofs of Key Steps for Exercise 2.D
Suppose that is Lindelof and that
is compact. Let
be an open cover of
. For each
, let
be finite such that
is a cover of
. Putting it another way,
. By the Tube lemma, for each
, there is an open
such that
. Since
is Lindelof, there exists a countable set
such that
is a cover of
. Then
is a countable subcover of
. This completes the proof that
is Lindelof when
is Lindelof and
is compact.
To complete the exercise, observe that if is Lindelof and
is
-compact, then
is the union of countably many Lindelof subspaces.
Proofs of Key Steps for Exercise 2.E
Proof of Fact E.1
Let be a compact Hausdorff space. Let
be a sequence of non-empty open subsets of
such that $latex
for each
. Show that the intersection
is non-empty.
Suppose that . Choose
. There must exist some
such that
. Choose
. There must exist some
such that
. Continue in this manner we can choose inductively an infinite set
such that
for
. Since
is compact, the infinite set
has a limit point
. This means that every open set containing
contains some
(in fact for infinitely many
). The point
cannot be in the intersection
. Thus for some
,
. Thus
. We can choose an open set
such that
and
. However,
must contain some point
where
. This is a contradiction since
for all
. Thus Fact E.1 is established.
Proof of Fact E.2
Let be a compact space. Let
be open subsets of
such that each
is also a dense subset of
. Let
a non-empty open subset of
. We wish to show that
contains a point that belongs to each
. Since
is dense in
,
is non-empty. Since
is dense in
, choose non-empty open
such that
and
. Since
is dense in
, choose non-empty open
such that
and
. Continue inductively in this manner and we have a sequence of open sets
just like in Fact E.1. Then the intersection of the open sets
is non-empty. Points in the intersection are in
and in all the
. This completes the proof of Fact E.2.
Proof of Fact E.3
Let be a Baire space. Let
be a dense
-subset of
such that
is a dense subset of
. Show that
is not a
-compact space.
Suppose is
-compact. Let
where each
is compact. Each
is obviously a closed subset of
. We claim that each
is a closed nowhere dense subset of
. To see this, let
be a non-empty open subset of
. Since
is dense in
,
contains a point
where
. Since
, there exists a non-empty open
such that
. This shows that each
is a nowhere dense subset of
.
Since is a dense
-subset of
,
where each
is an open and dense subset of
. Then each
is a closed nowhere dense subset of
. This means that
is the union of countably many closed and nowhere dense subsets of
. More specifically, we have the following.
(1)………
Statement (1) contradicts the fact that is a Baire space. Note that all
and
are open and dense subsets of
. Further note that the intersection of all these countably many open and dense subsets of
is empty according to (1). Thus
cannot not a
-compact space.
Proof of Fact E.4
The space is compact since it is a product of compact spaces. To see that
is a dense
-subset of
, note that
where for each integer
(2)………
Note that the first factors of
are the open interval
and the remaining factors are the closed interval
. It is also clear that
is a dense subset of
. This completes the proof of Fact E.4.
Concluding Remarks
Exercise 2.A
The exercise is to show that the product of uncountably many -compact spaces does not need to be Lindelof. The approach suggested in the hints is to show that
has uncountable extent where
is continuum. Having uncountable extent (i.e. having an uncountable subset that is both closed and discrete) implies the space is not Lindelof. The uncountable extent of the product space
is discussed in this post.
For and
, there is another way to show non-Lindelof. For example, both product spaces are not normal. As a result, both product spaces cannot be Lindelof. Note that every regular Lindelof space is normal. Both product spaces contain the product
as a closed subspace. The non-normality of
is discussed here.
Exercise 2.B
The hints given above is to show that the set of all irrational numbers, , is not
-compact (as a subspace of the real line). The same argument showing that
is not
-compact can be generalized. Note that the complement of
is
, the set of all rational numbers (a countable set). In this case,
is a dense subset of the real line and is the union of countably many singleton sets. Each singleton set is a closed and nowhere dense subset of the real line. In general, we can let
, the complement of a set
, be dense in the real line and be the union of countably many closed nowhere dense subsets of the real line (not necessarily singleton sets). The same argument will show that
cannot be a
-compact space. This argument is captured in Fact E.3 in Exercise 2.E. Thus both Exercise 2.B and Exercise 2.E use a Baire category argument.
Exercise 2.E
Like Exercise 2.B, this exercise is also to show a certain space is not -compact. In this case, the suggested space is
, the product of countably many copies of the real line. The hints given use a Baire category argument, as outlined in Fact E.1 through Fact E.4. The product space
is embedded in the compact space
, which is a Baire space. As mentioned earlier, Fact E.3 is essentially the same argument used for Exercise 2.B.
Using the same Baire category argument, it can be shown that , the product of countably many copies of the countably infinite discrete space, is not
-compact. The space
of the non-negative integers, as a subspace of the real line, is certainly
-compact. Using the same Baire category argument, we can see that the product of countably many copies of this discrete space is not
-compact. With the product space
, there is a connection with Exercise 2.B. The product
is homeomorphic to
. The idea of the homeomorphism is discussed here. Thus the non-
-compactness of
can be achieved by mapping it to the irrationals. Of course, the same Baire category argument runs through both exercises.
Exercise 2.C
Even the non--compactness of the Sorgenfrey line
can be achieved by a Baire category argument. The non-normality of the Sorgenfrey plane
can be achieved by Jones’ lemma argument or by the fact that
is not a first category set. Links to both arguments are given in the Proof section above.
See here for another introduction to the Baire category theorem.
The Tube lemma is discussed here.
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