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


In theorem 4.1 of ["Smoothing and extending cosmic time functions"](https://doi.org/10.1007/BF00759586 "H.-J. Seifert, Gen. Relat. Gravit. 8, 815–831 (1977). https://zbmath.org/?q=an:0425.53032") 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?"



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


