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  and .
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 .
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