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:



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> 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.
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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:



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> 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 *not* stably causal?
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Thank you for your kind help.



Subset of causal spacetime+Imprisonment Condition+Compact Closure -> Stably Causal spacetime?
*Edit:* In Beem's article on Page 180 before the introduction of condition (N) he mentions Carter's example (see also Hawking, Ellis The large scale structure of space-time, p. 195, Fig. 39). There you should get your counterexample by $E=(t\_1,t\_2)\times S^1 \times S^2$.


