A Quick Note About Lindelöf Spaces

Now that the semester is over, I actually have time to pay attention to this blog. Unfortunately I still have final exams to grade, but I really wanted to make a short entry on Lindelöf spaces. Part of the motivation here is that Lindelöf spaces are the source of many problems in general topology with (partial) set-theoretic solutions (Frank Tall has a nice paper on the subject).

Let us say that a Lindelöf space is X is productively Lindelöf if the product X\times Y is again Lindelöf for every Lindelöf space Y, and powerfully Lindelöf if its countable power X^\omega is again Lindelöf. We know of spaces which fail to even be productively Lindelöf with themselves. The Sorgenfrey line is a Lindelöf space such that the product with itself fails to be normal, and hence fails to be Lindelöf. One motivation for looking at these properties is that there is a natural class of spaces which are both productively and powerfully Lindelöf, the class of \sigma-compact spaces. Now, it turns out that under CH every “small enough” productively Lindelöf space is also powerfully Lindelöf, but not much is known outside of that.

Every time I’ve seen the result that the countable products of \sigma-compact spaces is Lindelöf, it’s been cited without proof as a folklore result. The only place I’ve seen a “proof” is by way of exercises in Engleking’s General Topology. The proof is somewhat indirect, and I was curious if there was a more direct way of showing this. I asked my topology professor, and he (off the top of his head) gave a very slick direct proof. Basically, I’m using this space to write down the argument so I don’t forget it.

Theorem: Let \{X_i : i\in \omega\} be a family of regular \sigma-compact spaces. Then the product \prod_{i\in\omega} X_i is Lindelöf.

Proof: We begin by writing expressing each X_i as the union of countably many compact sets, X_i=\bigcup_{n\in\omega} K_{n_i}. Now, consider the topological sum \bigsqcup_{n\in\omega} K_{n_i}, and note that there is a natural continuous onto map \bigsqcup_{n\in\omega} K_{n_i}\to X_i. Since the property of being Lindelöf is preserved by continuous images, it suffices to show that the product \prod_{i\in\omega}(\bigsqcup_{n\in\omega} K_{n_i}) is Lindelöf.

Next, we note that the space \bigsqcup_{n\in\omega} K_{n_i} consists of countably many disjoint clopen pieces. So, we can construct a continuous map \bigsqcup_{n\in\omega} K_{n_i}\to \omega by sending each K_{n_i}\mapsto n. In fact, this map is not only continuous, it is a perfect map (closed, onto, and point inverses are compact). Perfect maps have the following extremely nice properties:

  1. Preimages of Lindelöf spaces under perfect maps are again Lindelöf.
  2. The product of perfect maps is again perfect (this is a generalization of Tychonoff’s theorem that the product of compact spaces is compact).

So we see that this gives us a perfect map \prod_{i\in\omega}(\bigsqcup_{n\in\omega} K_{n_i})\to {}^\omega\omega. Since {}^\omega \omega is a seperable metric space, it follows that {}^\omega \omega is Lindelöf. This gives us that \prod_{i\in\omega}(\bigsqcup_{n\in\omega} K_{n_i}) is Lindelöf, and thus we can conclude that the product \prod_{i\in\omega} X_i is Lindelöf.

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