# Are there only countably many compact topological manifolds?

Up to homeomorphism, there are 2 one-dimensional topological manifolds and countably many 2- and 3-dimensional compact manifolds, respectively, since each manifold in these dimensions can be triangularized and hence be described by a finite amount of combinatorial data.

In higher dimensions this argument doesn't work anymore. So it still true for $n\ge 4$ that the set of homeomorphism classes of compact, connected topological $n$-manifolds (without boundary) is countable?

(I'd be also interested in the same question for diffeomorphism classes of compact smooth manifolds.)

• For diffeomorphism classes the answer is in the affirmative, as any smooth manifold can be triangulated and within each $PL$-class there are finitely many diffeomorphism classes. – Alex Degtyarev Feb 21 '15 at 11:53
• You also want to assume connectedness, since otherwise any disjoint union of closed manifolds would again be a closed manifold, and these disjoint unions can be arbitrarily big. – Tobias Fritz Feb 21 '15 at 13:18
• @TobiasFritz They would be finite disjoint unions. – Alex Degtyarev Feb 21 '15 at 13:35
• @Alex Degtyarev: can you give references? – ACL Feb 21 '15 at 14:55
• @Wlodzimierz: what do you mean exactly? – ACL Feb 21 '15 at 14:57