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.