What are the spacelike boundaries referred to in theorem 4.1 of "Smoothing and extending cosmic time functions" by Seifert

Question

I've been struggling with this question for a while.

In theorem 4.1 of "Smoothing and extending cosmic time functions" Seifert proves that a time function defined on a compact subset of a stably causal Lorentzian manifold is extendible to a global time function. To do this he constructs a countable collection of non-intersecting stable spacelike boundaries, $C_{\tau_k}$. The boundaries are inductively defined so that $$ C_{\tau_k}=\tilde{J}^-_{\theta_k}(Q_k)=\bigcup_{\eta>\theta_k}J^-(Q_k;g_{\eta}),$$ where $Q_k$ is some compact spacelike set, $\theta_k,\eta\in[a,b]$ for some $a,b\in\mathbb{R}$, $J^-(Q_k;g_{\eta})$ is the causal past of $Q_k$ with respect to the metric $g_{\eta}$ and for all $\alpha,\beta\in [a,b]$ we have that $g\lt g_\alpha$ and $g_\alpha\lt g_\beta$ if and only if $\alpha$ is less than $\beta$.

Seifert defines a boundary as a set $A$ so that there exists some $W\subset M$ so that $\partial W=A$. It seems to me that $I^-(C_{\tau_k},g)\subset C_{\tau_k}$ implying that $C_{\tau_k}$ has non-empty interior and therefore isn't a boundary. The set $\partial C_{\tau_k}$ seems to fit what Seifert wants, did he just forget a boundary symbol?

Since this paper was published in 1977 and has been cited roughly 15 times I would expect other researchers to comment on this problem. I've looked through the literature, however, and can't see anything that comments on this.

So my question is, "Is $C_{\tau_k}$ a boundary?"

Answer 1

Looking at Figure 3 and Step B of the proof of the theorem, it looks like the $C_{\tau_k}$ should be of the form $\partial \tilde{J}^-_{\theta_k}(Q_k)$. I am also pretty sure that he chose the symbol $C$ for "cone". Essentially the idea is to construct the surface as a union of a bunch of truncated cones that are wider than the causal cone for the given metric (so that the surface defined is automatically space-like).