If $x:\Lambda \rightarrow X$ is a net in a topological space $X$ and $\Lambda '\subseteq \Lambda$ is a cofinal subset of the directed set $\Lambda$, then $x|_{\Lambda '}$ is a subnet of $x$. We call subnets of this form *strict subnets*.

Of course, not all subnets are of this form, and indeed, the definition of a subnet is chosen the way it is in part so that certain theorems we would like to be true are in fact true. I am in particular curious about the following standard theorem.

$X$ is quasicompact iff every net in $X$ has a convergent subnet.

Looking at the proof, I definitely see where it would no go through if we were restricted to strict subnets, and it seems as if there would be no easy way to fix this with this restriction. Of course, this doesn't mean that it *can't* be proven---maybe we just haven't been clever enough to figure out how to do it with only strict subnets? I am quite confident this is not actually the case, however, and that there is indeed a counter-example.

Kelley presents an example (pg. 77 of his *General Topology*, attributed to Arens) that is meant to demonstrate why strict subnets are not enough. He presents an example of a net with a cluster point to which no subnet converges. Unfortunately, the example he gives is Lindelöf, but not quasicompact (and personally, I could live with nets not having subnets converging to cluster points (this is already the case for limit points, for example)).

So then, is there a well-known example of a quasicompact space and a net with no convergent *strict* subnet?