(Strong) causality condition

Question

I'm studying the causality in Lorentz Manifold with the book "Semi-Riemann Geometry", B.O'Neill. I have the followig problem: He says that, picked a subset A of a Lorentzian Manifold M, the causality condition hold on A if there aren't closed causal curve in A. After this he proves that if a set is compact it contains a closed causal curve. Then, if I understand, in a compact set the causality condition can't hold. But after this definition, the author proves a lemma in which he supposes that in a compact subset $K \subset M$ the causality condition holds. How can it be possible??

Answer 1

No closed casual curves $\implies$ Causality condition holds
Is equivalent to
Causality condition does not hold $\implies$ a closed casual curve exists

Not equivalent to
Closed casual curve exist $\color{red}\implies$ Causality condition doesn't hold

In your problem
A is compact $\implies$ a closed casual curve exists
this last condition does not imply that "Causality condition doesn't hold"