Stationary orbit space with its quotient topology

For the complete stationary flow ${\mathrm{\Phi }}_t$ on the exterior in the NT package, define $p\sim q$ when $q={\mathrm{\Phi }}_t(p)$ for some $t\in \mathbb{R}$. Let $Q=E\mathrm{/}\mathbb{R}$ be its set of orbits and let $\pi :E\to Q$ send each point to its orbit. Give $Q$ the quotient topology: $U\subset Q$ is open precisely when ${\pi }^{-1}(U)$ is open in $E$. A subset $K\subset Q$ is compact when every cover of it by open subsets of $Q$ has a finite subcover.

For a curve $\gamma :I\to E$, its stationary projection means $\pi \circ \gamma$, or its image $\pi (\gamma (I))$ when discussing containment. A compact set for the source NT condition is a compact subset of this open-exterior quotient, not a subset of a compactification containing the horizon or infinity. No choice of global time coordinate, smooth quotient structure, free action or proper action is required to state this topological condition; those additional source results are not assumed here.

The flow identities give reflexivity, symmetry and transitivity of $\sim$, so its equivalence classes define $Q$ and the surjection $\pi$. Preimages preserve arbitrary unions and finite intersections and take the empty set and whole space to their counterparts; hence the specified collection of subsets is a topology. The projection is continuous by this definition. The compactness predicate and curve-image containment therefore make sense without identifying $Q$ with a section or assuming a smooth-manifold theorem about it. In particular time drift along an orbit is removed by $\pi$, so boundedness in an arbitrary spacetime coordinate chart cannot replace this definition.