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Wikipedia defines a surface to be a two-dimensional manifold, and a closed surface to be a surface that is compact and without boundary. Am I correct that this definition of "closed surface" is not equivalent to the definition "a surface that is closed (as a Hausdorff space)"? A "closed surface" is defined to be compact, but a compact Hausdorff space is closed.

So, for example, a disk that includes its boundary is not a closed surface, but it is a surface that is closed (topologically). I find this definition very confusing.

Edit: As Joppy points out in their answer, this same issue applies to any closed manifold, not just two-dimensional surfaces.

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You're partly correct. A "closed manifold" is a manifold which is compact and has no boundary, and a "closed surface" is just a specialisation of that definition to dimension 2, and this is not equivalent to the definition of being closed as a Hausdorff space.

But any space is closed in itself, so every surface is topologically closed, so this is an uninteresting fact. This is why the re-use of the word "closed" is not too confusing.

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  • $\begingroup$ Every space is closed in itself, but not every subspace of Euclidean space is closed with respect to the inherited Euclidean topology. So I think the "closed-ness" of the manifold is trivial if you're thinking intrinsically, but nontrivial if you're thinking of it extrinsically, as being embedded in Euclidean space. $\endgroup$ – tparker Apr 23 '17 at 17:03
  • $\begingroup$ But thanks for pointing out that this same issue applies to any closed manifold. The Wiki article points out that "The notion of closed manifold is unrelated with that of a closed set. A disk with its boundary is a closed subset of the plane, but not a closed manifold." $\endgroup$ – tparker Apr 23 '17 at 17:11
  • $\begingroup$ @tparker But note that a closed manifold embedded in Euclidean space is compact and hence closed (and bounded) as a subspace. $\endgroup$ – Alex Provost Apr 23 '17 at 17:15
  • $\begingroup$ @AlexProvost True, closed as a manifold implies closed topologically, but not vice versa. $\endgroup$ – tparker Apr 23 '17 at 23:18

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