Compactness of the space of causal curves

TL;DR

This paper proves that the space of causal curves between compact sets in a certain class of posets is compact in the Vietoris topology, without relying on geometric or differentiable assumptions, extending results in general relativity.

Contribution

It establishes a purely order-theoretic proof of the compactness of causal curves space, bypassing geometric or differentiable structures.

Findings

The space of causal curves is compact in the Vietoris topology.

The result applies to separable globally hyperbolic posets.

The proof does not depend on geometric or differentiable assumptions.

Abstract

We prove that the space of causal curves between compact subsets of a separable globally hyperbolic poset is itself compact in the Vietoris topology. Although this result implies the usual result in general relativity, its proof does not require the use of geometry or differentiable structure.