# Characterizing compactness of the Alexandrov topology in a spacetime

This is perhaps more of a soft question and on the mathematical side of things, but I'm struggling to find references and to formulate a precise argument. There's of course the chance that what I'm about to ask turns out to be completely trivial, but let's see.

Whenever $$(M,\mathtt{g}$$) is a spacetime, we may consider its Alexandrov topology: the topology generated by the collection $$\{I^+(x) \cap I^-(y) \mid x,y\in M\}$$ of all chronological diamonds in $$(M,\mathtt{g})$$. It is always contained in the original manifold topology of $$M$$.

Here's a bunch of facts which are trivial or well-known:

1. the Alexandrov topology is always connected.
2. the Alexandrov topology is the chaotic topology $$\{\varnothing, M\}$$ if and only if $$(M,\mathtt{g})$$ is totally vicious.
3. the Alexandrov topology is Hausdorff if and only if $$(M,\mathtt{g})$$ is strongly causal (in which case the Alexandrov topology coincides with the manifold topology)
4. if the Alexandrov topology is compact, then $$(M,\mathtt{g})$$ is not chronological.

I would like to know if there's anything similar to a "converse" to item 4. While I understand that compactness of the causal diamonds is what really matters when progressing through the causal hierarchy (to obtain things like the Avez-Seifert theorem for globally hyperbolic spacetimes), I am still wondering if there is a simple characterization for the compactness of the Alexandrov topology.

I don't quite believe it's true that the converse of item 4. holds the way it is written now. If the space is not totally vicious and $$\gamma$$ is a closed timelike curve, the future $$I^+[\gamma]$$ may not be so big in $$M$$. I only know of a '84 result by Galloway essentially stating that if a compact spacetime $$(M,\mathtt{g})$$ satisfies $${\rm Ric}_x(v,v)>0$$ for every $$x\in M$$ and causal vector $$v\in T_xM$$, then $$(M,\mathtt{g})$$ is totally vicious (but I haven't studied the proof in detail to decide if compactness of $$M$$ may be replaced with compactness of the Alexandrov topology, and assuming such an energy condition is probably overkill).

I'll appreciate any comments. Thank you!

Consider a 2D Minkowski spacetime and identify two constant time surfaces $$S_T\equiv S_{-T}$$ for some $$T>0$$. Remove the lines $$t=0$$, $$x\geq 1$$ and $$t=0$$, $$x\leq -1$$ from the obtained cylinder. The arising spacetime is not chronological nor totally vicious. I think that it is not compact because the Alexandrov topology in this spacetime behaves like the Alexandrov topology in Minkowski spacetime (which is not compact because it coincides with the standard topology!) when considering couples of events, e.g. in the region $$0, $$x >> 1$$.