My question arose after studying the article "John K. Beem: Conformal Changes and Geodesic Completeness". (http://projecteuclid.org/euclid.cmp/1103899983) One of the results there is:

Let $(M,g)$ be a causal spacetime which satisfies condition $N$. If $E$ is an open subset of $M$ with compact closure $\overline{E}$, then $(E,g)$ is stably causal.

To recall the necessary definitions:

For $(M,g)$ condition $N$ is satisfied, if for every compact subset $K \subset M$, there is no future inextendible nonspacelike curve which is totally future imprisoned in $K$.

$(E,g)$ is the metric $g$ restricted to the manifold $E\subset M$.

However I feel the proof of this is flawed, (The negation of stable causality is stated falsely in the proof of theorem 2, I think) yet the claim seems to be true. In fact even ignoring the condition $N$, I could not come up with a counterexample. Can anyone provide me with a counterexample? That is to answer the following question:

Can there be a causal spacetime $(M,g)$ and an open subset $E \subset M$ with compact closure $\overline{E}$, such that $(E,g)$ is

notstably causal?

Thank you for your kind help.