Talk:Gibbons–Hawking–York boundary term

In the second paragraph "...is appropriate only when the underlying spacetime manifold M {\displaystyle {\mathcal {M}}} {\mathcal {M}} is closed, i.e., a manifold which is both compact and without boundary."

This is NOT the definition of closed (see Marsden and Hoffman). For subsets of R^n with a Euclidian distance function, closed + bounded ==> compact (Bolzano Weierstrauss theorem, compact is sequentially compact for metric spaces).

The only compact set without boundary is the empty set, and this is a vacuous example. Any compact subset of R^n necessarily contains a non-empty boundary: the closure of the set intersect the closure of its complement. The closure is the set union all its limit points. Limit points are all points accessible via cauchy convergence of sequences within the set. Consider [0,2]. The closure of this set is [0,2]. Its complement is (2, +inf) union (-inf, 0). The closure of the complement is [2, +inf) union (-inf, 0]. The intersection of these two closures is {0,2}, which is obviously the "edge" of [0,2].

So the characterization of M given here is badly flawed. Can someone fix it? 2607:F278:410E:5:9EB6:D0FF:FEF3:F2F1 (talk) 19:23, 13 August 2018 (UTC)

You are referring to the notion a closed subset of Euclidean space, or more generally of a topological space. The notion here is of a closed manifold. Gumshoe2 (talk) 03:01, 4 October 2020 (UTC)