Two footnotes in a paper of E. Michael

Some authors of papers introduce or motivate their results by giving basic facts either in the body of the papers or in footnotes. The basic information can be a treasure trove of information for those who study or review topology. In this post I discuss two footnotes in a paper of E. Michael (see [2]) and how they relate to the results/examples in the paper. Here’s the footnote 2 and footnote 4 in [2]:

  • Footnote 2: The reader should recall that paracompact spaces are normal, and that regular Lindelof spaces are paracompact. A Lindelof space X with all open subsets F_\sigma is hereditarily Lindelof, and conversely if X is regular.
  • Footnote 4: This example is new for n=1. In contrast to this example, S. Willard has shown that, if X \times Y is paracompact with X Lindelof and Y separable, then X \times Y must be Lindelof.

The title of the paper is Paracompactness and the Lindelof Property in Finite and Countable Cartesian Products. The example of the Sorgenfrey Line shows that the product of two Lindelof spaces needs not be normal. The goal of [2] is to present several examples demonstrating that higher powers of paracompact or Lindelof spaces can behave unpredictably too.

Footnote 2 gives background information about paracompact, Lindelof spaces, and hereditarily Lindelof spaces. The last sentence in the footnote is that any Lindelof space is hereditarily Lindelof if and only if it is perfectly normal. Footnote 4 provides some contrasting information to Example 1.4 in [2]. In the following discussion, all spaces are assumed to be Hausdorff and regular.

Discussion of Footnote 2
The last sentence in the footnote is essentially the following theorem:

Theorem 1
For any Lindelof space X, the space X is hereditarily Lindelof property if and only if X is perfectly normal.

A space is perfectly normal if it is normal and that every open subspace is an F_\sigma set. Thus, an alternative way to check whether a Lindelof space is hereditarily Lindelof is to check whether every open subset is F_\sigma (or every closed subset is G_\delta). In particular, any compact space with a closed subset (or even a singleton set) that is not G_\delta cannot be hereditarily Lindelof. Some examples: the unit square with the lexicographic order, the ordinal \omega_1+1, and [0,1]^{\mathcal{K}} where \mathcal{K} is any uncountable cardinal.

To prove Theorem 1, we need the following proposition.

Proposition 1
Any space X is hereditarily Lindelof if and only if every open subspace of X is Lindelof.

Proof. The direction \Rightarrow is clear. To see \Leftarrow, let Y \subset X and let \mathcal{U} be an open cover of Y. Let \mathcal{U}^* be a collection of open subsets of X such that for each U^* \in \mathcal{U}^*, U^* \cap Y=U for some U \in \mathcal{U}. Then \bigcup \mathcal{U}^* is Lindelof. We can find countably many sets in \mathcal{U}^* whose union equals \bigcup \mathcal{U}^*. It follows that we can find a countable subcollection of \mathcal{U} that covers Y.

Proof of Theorem 1. \Rightarrow Suppose X is hereditarily Lindelof. The normality of X comes from the fact that X is regular and Lindelof. Let U \subset X be an open subset. For each x \in U, let V_x be open such that x \in V_x \subset \overline{V_x} \subset U (this comes from the fact that X is a regular space). Since U is Lindelof, we can find countably many V_x such that the union of these countably many \overline{V_x} equals U. This shows that every open subset of X is an F_\sigma set.

\Leftarrow Suppose the Lindelof space X is perfectly normal. To show that X is hereditarily Lindelof, it suffices to show that every open subset of X is Lindelof. This follows from that fact that every open subset of X is an F_\sigma set and that the Lindelof property is hereditary with respect to F_\sigma subsets.

Discussion of Footnote 4
Example 1.4 in [2] provides, under the Continuum Hypothesis, for each positive integer n, a regular space Y such that Y^n is Lindelof and Y^{n+1} is paracompact, but Y^{n+1} is not Lindelof. For n=1, Example 1.4 is essentially a negative answer to the question: if X \times Y is paracompact and each of the factors is Lindelof, must X \times Y be Lindelof? Footnote 4 in [2] says that if one of the factors is separable, then X \times Y must be Lindelof. We have the following theorem.

Theorem 2
If X \times Y is paracompact such that X is Lindelof anf Y is separable, then X \times Y is Lindelof.

To prove Theorem 2, we need the following two results.

Theorem 3
If X is paracompact and has a dense Lindelof subspace, then X must be Lindelof.

Proof. Suppose that X is paracompact. Let Y \subset X be a dense Lindelof subspace. To show that X is Lindelof, it suffices to show that every locally finite open cover of X has a countable subcover.

Let \mathcal{U} be a locally finite open cover of X. For each y \in Y, choose open O_y such that y \in O_y and O_y only meets finitely many sets in \mathcal{U}. For each such O_y, let U_y be the union of the finitely many U \in \mathcal{U} that intersect O_y.

Since Y is Lindelof, we can find countably many O_y whose union contains Y. Consider the countably many corresponding U_y. We claim that the countably many U \in \mathcal{U} that are associated with these countably many U_y form a cover of X. Let x \in X. Choose some U \in \mathcal{U} such that x \in U. Since Y is dense in X, choose some z \in U \cap Y. Then z belongs to one of the countably many O_y that cover Y. Thus, U is associated with one of the corresponding U_y. This completes the proof of Theorem 3.

Theorem 4
If X is Lindelof and Y is a \sigma \text{-}compact space, then X \times Y is Lindelof.

Proof. It is known that X \times Y is Lindelof if one factor is Lindelof and the other factor is compact (see this previous post). As a corollary, if one of the factor is the union of countably many compact spaces, X \times Y is Lindelof.

Proof of Theorem 2. Suppose that X \times Y is paracompact and that X is Lindelof and Y is separable. Let D be a countable dense subset of Y. Then X \times D is Lindelof by Theorem 4. Furthermore, X \times D is a dense Lindelof subspace of X \times Y, By Theorem 3, X \times Y must be Lindelof.

Reference

  1. Engelking, R., General Topology, Revised and Completed edition, 1989, Heldermann Verlag, Berlin.
  2. Michael, E., Paracompactness and the Lindelof property in Finite and Countable Cartesian Products, Compositio Math. 23 (1971) 199-214.
  3. Willard, S., General Topology, 1970, Addison-Wesley Publishing Company.
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