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 -compact space with any Lindelof space is Lindelof. Another way to state this basic topological fact is that -compact spaces are productively Lindelof. We present an example of a productively Lindelof space that is not -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 -compact space with any Lindelof space is Lindelof is due to the Tube Lemma.

We now define a productively Lindelof space that is not -compact. Let where is any uncountable set and . The set is discrete in and open neighborhoods at have the form where is countable. The only compact subsets of this space are finite sets. Thus is not -compact.

To see that is productively Lindelof, let be any Lindelof space. Let be any open cover of . Assume that consists of open sets of the form where is open in and is open in .

There exists a countable such that covers . Suppose that . Also assume that for each , where is countable.

Note that each is a Lindelof space since it is the product of a countable space (thus -compact) with a Lindelof space. It is also clear that each point either belongs to a set in or to for some .

For each , choose countable such that covers . Then is a countable subcover of .

*Reference*

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

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