Every space is star discrete

The statement in the title is a folklore fact, though the term star discrete is usually not used whenever this well known fact is invoked in the literature. We present a proof to this well known fact. We also discuss some related concepts.

All spaces are assumed to be Hausdorff and regular.

First, let’s define the star notation. Let X be a space. Let \mathcal{U} be a collection of subsets of X. Let A \subset X. Define \text{St}(A,\mathcal{U}) to be the set \bigcup \{U \in \mathcal{U}: U \cap A \ne \varnothing \}. In other words, the set \text{St}(A,\mathcal{U}) is simply the union of all elements of \mathcal{U} that contains points of the set A. The set \text{St}(A,\mathcal{U}) is also called the star of the set A with respect to the collection \mathcal{U}. If A=\{ x \}, we use the notation \text{St}(x,\mathcal{U}) instead of \text{St}( \{ x \},\mathcal{U}). The following is the well known result in question.

Lemma 1
Let X be a space. For any open cover \mathcal{U} of X, there exists a discrete subspace A of X such that X=\text{St}(A,\mathcal{U}). Furthermore, the set A can be chosen in such a way that it is also a closed subset of the space X.

Any space that satisfies the condition in Lemma 1 is said to be a star discrete space. The proof shown below will work for any topological space. Hence every space is star discrete. We come across three references in which the lemma is stated or is used – Lemma IV.2.20 in page 135 of [3], page 137 of [2] and [1]. The first two references do not use the term star discrete. Star discrete is mentioned in [1] since that paper focuses on star properties. This property that is present in every topological space is at heart a covering property. Here’s a rewording of the lemma that makes it look like a covering property.

Lemma 1a
Let X be a space. For any open cover \mathcal{U} of X, there exists a discrete subspace A of X such that \{ \text{St}(x,\mathcal{U}): x \in A \} is a cover of X. Furthermore, the set A can be chosen in such a way that it is also a closed subset of the space X.

Lemma 1a is clearly identical to Lemma 1. However, Lemma 1a makes it extra clear that this is a covering property. For every open cover of a space, instead of finding a sub cover or an open refinement, we find a discrete subspace so that the stars of the points of the discrete subspace with respect to the given open cover also cover the space.

Lemma 1a naturally leads to other star covering properties. For example, a space X is said to be a star countable space if for any open cover \mathcal{U} of X, there exists a countable subspace A of X such that \{ \text{St}(x,\mathcal{U}): x \in A \} is a cover of X. A space X is said to be a star Lindelof space if for any open cover \mathcal{U} of X, there exists a Lindelof subspace A of X such that \{ \text{St}(x,\mathcal{U}): x \in A \} is a cover of X. In general, for any topological property \mathcal{P}, a space X is a star \mathcal{P} space if for any open cover \mathcal{U} of X, there exists a subspace A of X with property \mathcal{P} such that \{ \text{St}(x,\mathcal{U}): x \in A \} is a cover of X.

It follows that every Lindelof space is a star countable space. It is also clear that every star countable space is a star Lindelof space.

Lemma 1 or Lemma 1a, at first glance, may seem like a surprising result. However, one can argue that it is not a strong result at all since the property is possessed by every space. Indeed, the lemma has nothing to say about the size of the discrete set. It only says that there exists a star cover based on a discrete set for a given open cover. To derive more information about the given space, we may need to work with more information on the space in question.

Consider spaces such that every discrete subspace is countable (such a space is said to have countable spread or a space of countable spread). Also consider spaces such that every closed and discrete subspace is countable (such a space is said to have countable extent or a space of countable extent). Any space that has countable spread is also a space that has countable extent for the simple reason that if every discrete subspace is countable, then every closed and discrete subspace is countable.

Then it follows from Lemma 1 that any space X that has countable extent is star countable. Any star countable space is obviously a star Lindelof space. The following diagram displays these relationships.

Countable spread and Lindelof property

According to the diagram, the star countable and star Lindelof are both downstream from the countable spread property and the Lindelof property. The star properties being downstream from the Lindelof property is not surprising. What is interesting is that if a space has countable spread, then it is star countable and hence star Lindelof.

Do “countable spread” and “Lindelof” relate to each other? Lindelof spaces do not have to have countable spread. The simplest example is the one-point compactification of an uncountable discrete space. More specifically, let X be an uncountable discrete space. Let p be a point not in X. Then Y=X \cup \{ p \} is a compact space (hence Lindelof) where X is discrete and an open neighborhood of p is of the form \{ p \} \cup U where X-U is a finite subset of X. The space Y is not of countable spread since X is an uncountable discrete subspace.

Does “countable spread” imply “Lindelof”? Is there a non-Lindelof space that has countable spread? It turns out that the answers are independent of ZFC. The next post has more details.

We now give a proof to Lemma 1. Suppose that X is an infinite space (if it is finite, the lemma is true since the space is Hausdorff). Let \kappa=\lvert X \lvert. Let \kappa^+ be the next cardinal greater than \kappa. Let \mathcal{U} be an open cover of the space X. Choose x_0 \in X. We choose a sequence of points x_0,x_1,\cdots,x_\alpha,\cdots inductively. If \text{St}(\{x_\beta: \beta<\alpha \},\mathcal{U}) \ne X, we can choose a point x_\alpha \in X such that x_\alpha \notin \text{St}(\{x_\beta: \beta<\alpha \},\mathcal{U}).

We claim that the induction process must stop at some \alpha<\kappa^+. In other words, at some \alpha<\kappa^+, the star of the previous points must be the entire space and we run out of points to choose. Otherwise, we would have obtained a subset of X with cardinality \kappa^+, a contradiction. Choose the least \alpha<\kappa^+ such that \text{St}(\{x_\beta: \beta<\alpha \},\mathcal{U}) = X. Let A=\{x_\beta: \beta<\alpha \}.

Then it can be verified that the set A is a discrete subspace of X and that A is a closed subset of X. Note that x_\beta \in \text{St}(x_\beta, \mathcal{U}) while x_\gamma \notin \text{St}(x_\beta, \mathcal{U}) for all \gamma \ne \beta. This follows from the way the points are chosen in the induction process. On the other hand, for any x \in X-A, x \in \text{St}(x_\beta, \mathcal{U}) for some \beta<\alpha. As discussed, the open set \text{St}(x_\beta, \mathcal{U}) contains only one point of A, namely x_\beta.

Reference

  1. Alas O., Jumqueira L., van Mill J., Tkachuk V., Wilson R.On the extent of star countable spaces, Cent. Eur. J. Math., 9 (3), 603-615, 2011.
  2. Alster, K., Pol, R.,On function spaces of compact subspaces of \Sigma-products of the real line, Fund. Math., 107, 35-46, 1980.
  3. Arkhangelskii, A. V.,Topological Function Spaces, Mathematics and Its Applications Series, Kluwer Academic Publishers, Dordrecht, 1992.

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