The weakly Lindelof property is a natural weakening of the familiar Lindelof property. In this post, we discuss some of the basic properties of weakly Lindelof spaces.
We consider topological spaces that are at least (i.e. finite sets are closed) and regular. A space is said to be Lindelof if for any open cover of , there is a countable such that . A natural weakening of the Lindelof property is that we only require the countable to cover a dense subset of the space . Specifically, a space is said to be a weakly Lindelof space if for any open cover of , there is a countable such that is dense in .
The notion of weakly Lindelof has a brief mention in the Encyclopedia of General Topology (see page 183 in ), pointing out a connection to Banach space theory. Furthermore, assuming CH, the weakly Lindelof subspaces of are precisely those subspaces which are -embedded into . In this post, we focus on the basic properties.
Clearly separable spaces and Lindelof spaces are weakly Lindelof. Another obvious property that implies weakly Lindelof is the existence of a dense Lindelof subspace. It is slightly less obvious that the countable chain condition implies the weakly Lindelof property. We have the following implications.
All the affirmative implications in the above diagram cannot be reversed (see Examples 1, 2 and 3 below).
Some Cardinal Functions
Some of the properties discussed below can be described by cardinal functions, e.g., Lindelof number and weak Lindelof numbers. So we describe these before going into the basic properties. Let be a space. The Lindelof number of the space , denoted by , is the least cardinal number such that every open cover of has a subcollection with such that is a cover of . When , we say that the space is Lindelof.
The weak Lindelof number of the space , denoted by , is the least cardinal number such that every open cover of has a subcollection with such that . When , we say that the space is weakly Lindelof.
The character at , denoted by , is the least cardinal number of a local base at the point . The character of the space , denoted by , is the supremum of all the cardinal numbers over all . When , we say that is first countable.
The cellularity of the space , denoted by , is the least infinite cardinal number such that every collection of pairwise disjoint non-empty open subsets of has cardinality . When , we say that has the countable chain condition.
The extent of the space , denoted by , is the least infinite cardinal number such that if is a closed and discrete subset of , then . If , then is said to have countable extent (there are no uncountable closed and discrete subset). It is well known that Lindelof spaces have countable extent. The Lindelof number and the extent is related by the inequality: .
Weakly Lindelof spaces behave differently from Lindelof spaces in some ways. For example, closed subsets of a weakly Lindelof space do not have to be weakly Lindelof. In other ways, weakly Lindelof spaces and Lindelof spaces behave similarly. For example, the product of weakly Lindelof spaces needs not be weakly Lindelof and that every continuous image of a weakly Lindelof space is weakly Lindelof. Any Lindelof, Hausdorff and first countable space has cardinality no more than continuum. There is a similar theorem for weakly Lindelof spaces. Despite all these similarities with Lindelof spaces, the weak Lindelof property is a very weak property. It is well known that every Lindelof space has countable extent. There is no bound on the extent of weakly Lindelof spaces. The extent of a weakly Lindelof space can be arbitrarily large (see Example 4 below).
We discuss the following properties of weakly Lindelof spaces.
- Any space with the countable chain condition is weakly Lindelof.
- Any paracompact weakly Lindelof space is Lindelof.
- Every continuous image of a weakly Lindelof space is weakly Lindelof.
- The product of a compact space and a weakly lindelof space is weakly Lindelof.
- The product of two Lindelof spaces needs not be weakly Lindelof.
- Any normal first countable weakly Lindelof space has cardinality .
- For any infinite cardinal , there exists a weakly Lindelof space such that , i.e., the extent is at least . See Example 4 below.
Proof of 1
A space has the countable chain condition (has the CCC or is CCC for short) if there exists no uncountable collection of non-empty pairwise disjoint open subsets of . “CCC weakly Lindelof” follows from the following theorem (proved in this previous post).
A space has the CCC if and only if for every collection of non-empty open subsets of , there is a countable such that .
To finish off, let be an open cover of . By the theorem, there exists a countable such that . This means that .
Even though CCC implies weakly Lindelof, CCC does not imply the stronger property of having a dense Lindelof subspace (see Example 3 below).
The proof of 1 can be generalized to show that for any space . However, the inequality cannot be made an equality. In fact, the inequality can be made as wide as one wishes. Specifically, we can keep while making as large as one wishes (see Example 2 below). Thus the notions of countable chain condition and the weakly Lindelof property are far apart.
Proof of 2
Let be an open cover of a paracompact weakly Lindelof space . Using the regularity of the space, there is an open refinement of for each , for some . Using the paracompactness, let be a locally finite open refinement of . Using the weakly Lindelof property, choose a countable such that . With the collection being locally finite, we have . Thus every point of belongs to some for some . Tracing from to , to and then to , we see that for every , for some . It follows that a countable subcollection of is a cover of . This completes the proof of bullet point 2.
This result implies that in any metrizable space, the weakly Lindelof number coincides with the Lindelof number. So in metrizable spaces, the weak Lindelof number is just as good as an indicator of weight as the other cardinal functions such as density and Lindelof number.
Among CCC spaces, paracompactness and the Lindelof property coincide. This result shows that among weakly Lindelof spaces, paracompactness and the Lindelof property also coincide.
The proof of 3 is straightforward. It is very similar to the proof that continuous image of a Lindelof space is Lindelof.
Proof of 4
The proof that the product of a compact space and a weakly Lindelof space is weakly Lindelof makes use of the tube lemma, as in the proof that the product of a compact space and a Lindelof space is Lindelof.
Let be weakly Lindelof. Let be compact. Let be an open cover of . For each , let be finite such that is a cover of . By the tube lemma, there exists some open set such that .
Since is weakly Lindelof, there exists a countable such that . Let . It is clear that is a countable subcollection of . Note that the set is dense in . Thus the set is dense in too. Thus . This completes the proof that is weakly Lindelof.
Proof of 5
An example of two Lindelof spaces whose product is not weakly Lindelof is provided in .
Discussion of 6
Any Lindelof first countable Hausdorff space has cardinality no more than continuum (discussed in this previous post). This fact is a specific case of the general theorem that
for any Hausdorff space . Hence, the cardinality of any first countable Lindelof space is bounded by . It is interesting that there is an analogous result for weakly Lindelof space. In , the following inequality was proved:
for any normal space (Theorem 2.1 in ). Thus the cardinality of any normal weakly Lindelof space is bounded by .
Example 1 and Example 2 below use Lindelof or compact spaces that do not have the CCC as starting point. Here’s several examples of Lindelof non-CCC spaces:
- One-point Lindelofication of an uncountable set. The space is denoted by and is the set where is the discrete space of cardinality and is a point not in . The open neighborhoods at have the form where is countable.
- The space with the order topology. Note that is the immediate successor of , the first uncountable ordinal. See here.
- The unit square with the lexicographic order. See here.
- The Alexandroff Double Circle. See here.
In the above four spaces, the first one is Lindelof and the other three are compact. All four do not have the countable chain condition.
A non-Lindelof space that has a dense Lindelof subspace. As a bonus, this space does not have the CCC.
The idea is to start with a space that has a countable dense set of isolated points and an uncountable closed and discrete subset. One such space is a so called psi-space, a space defined using an uncountable almost disjoint family of subsets of . Then replace each of the countably many isolated points with a copy of one of the above examples of a Lindelof space without the CCC.
Let the first infinite ordinal (or the set of all nonnegative integers). Let be an uncountable almost disjoint family of subsets of (for the purpose of this example, it does not have to be an maximal almost disjoint family). Let , where each is isolated and each has open neighborhoods of the form where is finite. For a more detailed discussion about Psi-space, see this previous post.
Let be any one of the above Lindelof space that is not CCC. For each , let . So the are distinct copies of the space . The underlying set of this example is the following set:
The topology on is defined in such a way that each is considered a copy of the space and each has open neighborhoods of the form:
where is finite. The union of all is a dense Lindelof subspace of . The set is an uncountable closed and discrete subset of . Thus is not Lindelof. Each has uncountably many disjoint open sets. Thus does not have the CCC. This example shows that the existence of a dense Lindelof subspace implies neither the CCC nor the Lindelof property.
A weakly Lindelof non-CCC space .
Let be any one of the above three non-CCC compact spaces. Let be any space with the CCC, hence is weakly Lindelof. Let . Then is weakly Lindelof. It is also clear that does not have the CCC. This example shows that the weakly Lindelof property does not imply the countable chain condition.
This example shows that . In fact, it is possible to make as large as possible. In the definition of in this example, let be the one-point Lindelofication and be any CCC space. Then can be made as large as possible. Hence can be made as large as possible.
A CCC space that has no dense Lindelof subspace.
This example is found in a paper of Arhangel’skii (Theorem 1.1 in ). Let be the set of all continuous real-valued functions defined on . The set endowed with the pointwise convergence topology is typically denoted by . The space we want to use is .
The space is a dense subspace of the product space . Thus has the CCC. In , it is shown that does not contain a dense normal subspace. Hence it does not contain a dense Lindelof subspace. The proof that does not contain a dense normal subspace is a deep and non-trivial result.
The example shows that even though CCC implies the weakly Lindelof property, it cannot give the stronger property of the existence of a dense Lindelof subspace. It is also an example showing that the implication “existence of a dense Lindelof subspace weakly Lindelof” cannot be reversed.
An weakly Lindelof space such that the extent can be made arbitrarily large.
Let be any uncountable cardinal. Let be a discrete space of cardinality . Let be the Stone-Cech compactification of . Consider the ordinal with the order topology (can just think of it as a sequence of isolated points converging to the limit ). The space is defined as follows:
Note that is a -compact dense subspace of . Hence is weakly Lindelof. On the other hand, the set is a closed and discrete subset of . Since the cardinality of can be made arbitrarily large, the extent of can be made arbitrarily large. Thus there is no upper bound on the extent of weakly Lindelof spaces (unlike Lindelof spaces).
- Arhangel’skii A. V., Normality and Dense Subspaces, Proc. Amer. Math. Soc., 48, no. 2, 283-291, 2001.
- Bell M., Ginsburg J., Woods G., Cardinal Inequalities for Topological Spaces Involving the Weak Lindelof Number, Pacific J. Math., 79, no. 1, 37-45, 1978.
- Hajnal A., Juhasz I., On the Products of Weakly Lindelof Spaces, Proc. Amer. Math. Soc., 130, no. 1, 454-456, 1975.
- Hart, K. P., Nagata J. I., Vaughan, J. E., editors, Encyclopedia of General Topology, First Edition, Elsevier Science Publishers B. V, Amsterdam, 2003.