Weakly Lindelof spaces

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 T_1 (i.e. finite sets are closed) and regular. A space X is said to be Lindelof if for any open cover \mathcal{U} of X, there is a countable \mathcal{V} \subset \mathcal{U} such that X=\bigcup \mathcal{V}. A natural weakening of the Lindelof property is that we only require the countable \mathcal{V} to cover a dense subset of the space X. Specifically, a space X is said to be a weakly Lindelof space if for any open cover \mathcal{U} of X, there is a countable \mathcal{V} \subset \mathcal{U} such that \bigcup \mathcal{V} is dense in X.

The notion of weakly Lindelof has a brief mention in the Encyclopedia of General Topology (see page 183 in [4]), pointing out a connection to Banach space theory. Furthermore, assuming CH, the weakly Lindelof subspaces of \beta \mathbb{N} are precisely those subspaces which are C^*-embedded into \beta \mathbb{N}. 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).

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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 X be a space. The Lindelof number of the space X, denoted by L(X), is the least cardinal number \mathcal{K} such that every open cover \mathcal{U} of X has a subcollection \mathcal{V} \subset \mathcal{U} with \lvert \mathcal{V} \lvert \le \mathcal{K} such that \mathcal{V} is a cover of X. When L(X)=\omega, we say that the space is Lindelof.

The weak Lindelof number of the space X, denoted by wL(X), is the least cardinal number \mathcal{K} such that every open cover \mathcal{U} of X has a subcollection \mathcal{V} \subset \mathcal{U} with \lvert \mathcal{V} \lvert \le \mathcal{K} such that X=\overline{\bigcup \mathcal{V}}. When wL(X)=\omega, we say that the space is weakly Lindelof.

The character at x \in X, denoted by \chi(x,X), is the least cardinal number of a local base at the point x \in X. The character of the space X, denoted by \chi(X), is the supremum of all the cardinal numbers \chi(x,X) over all x \in X. When \chi(X)=\omega, we say that X is first countable.

The cellularity of the space X, denoted by c(X), is the least infinite cardinal number \mathcal{K} such that every collection of pairwise disjoint non-empty open subsets of X has cardinality \le \mathcal{K}. When c(X)=\omega, we say that X has the countable chain condition.

The extent of the space X, denoted by e(X), is the least infinite cardinal number \mathcal{K} such that if A is a closed and discrete subset of X, then \lvert A \lvert \le \mathcal{K}. If e(X)=\omega, then X 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: e(X) \le L(X).

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Basic Properties

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.

  1. Any space with the countable chain condition is weakly Lindelof.
  2. Any paracompact weakly Lindelof space is Lindelof.
  3. Every continuous image of a weakly Lindelof space is weakly Lindelof.
  4. The product of a compact space and a weakly lindelof space is weakly Lindelof.
  5. The product of two Lindelof spaces needs not be weakly Lindelof.
  6. Any normal first countable weakly Lindelof space has cardinality \le 2^\omega.
  7. For any infinite cardinal \mathcal{K}, there exists a weakly Lindelof space X such that e(X) \ge \mathcal{K}, i.e., the extent is at least \mathcal{K}. See Example 4 below.

Proof of 1
A space X 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 X. “CCC \Longrightarrow weakly Lindelof” follows from the following theorem (proved in this previous post).

    Theorem
    A space X has the CCC if and only if for every collection \mathcal{U} of non-empty open subsets of X, there is a countable \mathcal{V} \subset \mathcal{U} such that \bigcup \mathcal{U} \subset \overline{\bigcup \mathcal{V}}.

To finish off, let \mathcal{U} be an open cover of X. By the theorem, there exists a countable \mathcal{V} \subset \mathcal{U} such that \bigcup \mathcal{U} \subset \overline{\bigcup \mathcal{V}}. This means that X=\overline{\bigcup \mathcal{V}}. \blacksquare

Comments
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 wL(X) \le c(X) for any space X. However, the inequality cannot be made an equality. In fact, the inequality wL(X) \le c(X) can be made as wide as one wishes. Specifically, we can keep wL(X)=\omega while making c(X) 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 \mathcal{U} be an open cover of a paracompact weakly Lindelof space X. Using the regularity of the space, there is an open refinement \mathcal{V} of \mathcal{U} for each V \in \mathcal{V}, \overline{V} \subset U for some U \in \mathcal{U}. Using the paracompactness, let \mathcal{W} be a locally finite open refinement of \mathcal{V}. Using the weakly Lindelof property, choose a countable \mathcal{C} \subset \mathcal{W} such that X=\overline{\bigcup \mathcal{C}}. With the collection \mathcal{C} being locally finite, we have X=\overline{\bigcup \mathcal{C}}=\bigcup \left\{\overline{C}: C \in \mathcal{C} \right\}. Thus every point of X belongs to some \overline{C} for some C \in \mathcal{C}. Tracing from \mathcal{C} to \mathcal{W}, to \mathcal{V} and then to \mathcal{U}, we see that for every C \in \mathcal{C}, \overline{C} \subset U for some U \in \mathcal{U}. It follows that a countable subcollection of \mathcal{U} is a cover of X. 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. \blacksquare

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 X be weakly Lindelof. Let Y be compact. Let \mathcal{U} be an open cover of X \times Y. For each x \in X, let \mathcal{U}_x \subset \mathcal{U} be finite such that \mathcal{U}_x is a cover of \left\{ x \right\} \times Y. By the tube lemma, there exists some open set O_x \subset X such that \left\{ x \right\} \times Y \subset O_x \times Y \subset \bigcup \mathcal{U}_x.

Since X is weakly Lindelof, there exists a countable A \subset X such that X=\overline{\bigcup \limits_{x \in A} O_x}. Let \mathcal{U}_A=\bigcup \limits_{x \in A} \mathcal{U}_x. It is clear that \mathcal{U}_A is a countable subcollection of \mathcal{U}. Note that the set \bigcup \limits_{x \in A} (O_x \times Y) is dense in X \times Y. Thus the set \bigcup \bigcup \limits_{x \in A} \mathcal{U}_x is dense in X \times Y too. Thus X \times Y=\overline{\bigcup \bigcup \limits_{x \in A} \mathcal{U}_x}. This completes the proof that X \times Y is weakly Lindelof. \blacksquare

Proof of 5
An example of two Lindelof spaces whose product is not weakly Lindelof is provided in [3]. \blacksquare

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

    \lvert X \lvert \le 2^{\chi(X) \cdot L(X)}

for any Hausdorff space X. Hence, the cardinality of any first countable Lindelof space is bounded by 2^\omega. It is interesting that there is an analogous result for weakly Lindelof space. In [2], the following inequality was proved:

    \lvert X \lvert \le 2^{\chi(X) \cdot wL(X)}

for any normal space (Theorem 2.1 in [2]). Thus the cardinality of any normal weakly Lindelof space is bounded by 2^\omega.

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Examples

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 L(\mathcal{K}) and is the set \left\{p \right\} \cup D(\mathcal{K}) where D(\mathcal{K}) is the discrete space of cardinality \mathcal{K} and p is a point not in D(\mathcal{K}). The open neighborhoods at p have the form \left\{p \right\} \cup (D(\mathcal{K})-C) where C \subset D(\mathcal{K}) is countable.
  • The space \omega_1+1 with the order topology. Note that \omega_1+1 is the immediate successor of \omega_1, the first uncountable ordinal. See here.
  • The unit square [0,1] \times [0,1] 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.

Example 1
A non-Lindelof space X_1 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 \omega. 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 \omega the first infinite ordinal (or the set of all nonnegative integers). Let \mathcal{A} be an uncountable almost disjoint family of subsets of \omega (for the purpose of this example, it does not have to be an maximal almost disjoint family). Let \Psi(\mathcal{A})=\mathcal{A} \cup \omega, where each n \in \omega is isolated and each A \in \mathcal{A} has open neighborhoods of the form \left\{A \right\} \cup (A-F) where F \subset \omega is finite. For a more detailed discussion about Psi-space, see this previous post.

Let Y be any one of the above Lindelof space that is not CCC. For each n \in \omega, let Y_n=Y \times \left\{n \right\}. So the Y_n are distinct copies of the space Y. The underlying set of this example is the following set:

    X_1=\mathcal{A} \cup \bigcup \limits_{n \in \omega} Y_n

The topology on X_1 is defined in such a way that each Y_n is considered a copy of the space Y and each A \in \mathcal{A} has open neighborhoods of the form:

    \left\{A \right\} \cup \bigcup \limits_{n \in A-F} Y_n

where F \subset \omega is finite. The union of all Y_n is a dense Lindelof subspace of X_1. The set \mathcal{A} is an uncountable closed and discrete subset of X_1. Thus X_1 is not Lindelof. Each Y_n has uncountably many disjoint open sets. Thus X_1 does not have the CCC. This example shows that the existence of a dense Lindelof subspace implies neither the CCC nor the Lindelof property.

Example 2
A weakly Lindelof non-CCC space X_2.

Let X be any one of the above three non-CCC compact spaces. Let Y be any space with the CCC, hence is weakly Lindelof. Let X_2=X \times Y. Then X \times Y is weakly Lindelof. It is also clear that X \times Y does not have the CCC. This example shows that the weakly Lindelof property does not imply the countable chain condition.

This example shows that \omega=wL(X_2)<c(X_2). In fact, it is possible to make c(X_2) as large as possible. In the definition of X \times Y in this example, let X be the one-point Lindelofication L(\mathcal{K}) and Y be any CCC space. Then c(L(\mathcal{K})) can be made as large as possible. Hence c(X \times Y) can be made as large as possible.

Example 3
A CCC space X_3 that has no dense Lindelof subspace.

This example is found in a paper of Arhangel’skii (Theorem 1.1 in [1]). Let C(\omega_1+1) be the set of all continuous real-valued functions defined on \omega_1+1. The set C(\omega_1+1) endowed with the pointwise convergence topology is typically denoted by C_p(\omega_1+1). The space we want to use is X_3=C_p(\omega_1+1).

The space C_p(\omega_1+1) is a dense subspace of the product space \mathbb{R}^{\omega_1}. Thus C_p(\omega_1+1) has the CCC. In [1], it is shown that C_p(\omega_1+1) does not contain a dense normal subspace. Hence it does not contain a dense Lindelof subspace. The proof that C_p(\omega_1+1) does not contain a dense normal subspace is a deep and non-trivial result.

The example X_3=C_p(\omega_1+1) 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 \Longrightarrow weakly Lindelof” cannot be reversed.

Example 4
An weakly Lindelof space X_4 such that the extent can be made arbitrarily large.

Let \mathcal{K} be any uncountable cardinal. Let W be a discrete space of cardinality \mathcal{K}. Let \beta W be the Stone-Cech compactification of W. Consider the ordinal S=\omega+1 with the order topology (can just think of it as a sequence of isolated points converging to the limit \omega). The space X_4 is defined as follows:

    X_4=\beta W \times S-(\beta W-W) \times \left\{\omega \right\}

Note that \beta W \times \omega is a \sigma-compact dense subspace of X_4. Hence X_4 is weakly Lindelof. On the other hand, the set W \times \left\{\omega \right\} is a closed and discrete subset of X_4. Since the cardinality of W can be made arbitrarily large, the extent of X_4 can be made arbitrarily large. Thus there is no upper bound on the extent of weakly Lindelof spaces (unlike Lindelof spaces).

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Reference

  1. Arhangel’skii A. V., Normality and Dense Subspaces, Proc. Amer. Math. Soc., 48, no. 2, 283-291, 2001.
  2. 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.
  3. Hajnal A., Juhasz I., On the Products of Weakly Lindelof Spaces, Proc. Amer. Math. Soc., 130, no. 1, 454-456, 1975.
  4. Hart, K. P., Nagata J. I., Vaughan, J. E., editors, Encyclopedia of General Topology, First Edition, Elsevier Science Publishers B. V, Amsterdam, 2003.

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\copyright \ 2014 \text{ by Dan Ma}

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