An elementary example of a productively Lindelof space

A topological space is productively Lindelof if its product with every Lindelof space is Lindelof. It is well known that the product of a compact space with any Lindelof space is Lindelof. As a corollary, the product of a \sigma-compact space with any Lindelof space is Lindelof. Another way to state this basic topological fact is that \sigma-compact spaces are productively Lindelof. We present an example of a productively Lindelof space that is not \sigma-compact, demonstrating that these two notions are not equivalent. This example is an elementary one. No heavy machinery is required to define the example. References for productively Lindelof spaces include [1] and [3].

The fact that the product of any \sigma-compact space with any Lindelof space is Lindelof is due to the Tube Lemma.

We now define a productively Lindelof space that is not \sigma-compact. Let X= \left\{p\right\} \cup D where D is any uncountable set and p \notin D. The set D is discrete in X and open neighborhoods at p have the form \left\{p\right\} \cup (D-A) where A \subset D is countable. The only compact subsets of this space are finite sets. Thus X is not \sigma-compact.

To see that X is productively Lindelof, let Y be any Lindelof space. Let \mathcal{U} be any open cover of X \times Y. Assume that \mathcal{U} consists of open sets of the form G \times H where G is open in X and H is open in Y.

There exists a countable \mathcal{V} \subset \mathcal{U} such that \mathcal{V} covers \left\{p\right\} \times Y. Suppose that \mathcal{V}=\left\{G_1 \times H_1,G_2 \times H_2,G_3 \times H_3,\cdots \right\}. Also assume that for each i, G_i=\left\{p\right\} \cup (D-A_i) where A_i is countable.

Note that each A_i \times Y is a Lindelof space since it is the product of a countable space (thus \sigma-compact) with a Lindelof space. It is also clear that each point (x,y) \in X \times Y either belongs to a set in \mathcal{V} or to A_i \times Y for some i.

For each i, choose countable \mathcal{W}_i \subset \mathcal{U} such that \mathcal{W}_i covers A_i \times Y. Then \mathcal{V} \cup \mathcal{W}_1 \cup \mathcal{W}_2 \cup \cdots is a countable subcover of \mathcal{U}.

Reference

  1. Alster, K., On spaces whose product with every Lindelof space is Lindelof, Colloq. Math. 54 (1987), 171–178.
  2. Engelking, R., General Topology, Revised and Completed edition, 1989, Heldermann Verlag, Berlin.
  3. Tall, F., Productively Lindelof spaces may all be D, Canad. Math. Bull. to apear.
  4. Willard, S., General Topology, 1970, Addison-Wesley Publishing Company.
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