Definition talk:Loop (Topology)/Topological Manifold

Just being curious here: why does a loop in a topological manifold have to be on a continuous path, but a loop in a general topological space not have to be continuous? --prime mover (talk) 06:20, 16 December 2022 (UTC)

Actually, in books by Lee all loops are continuous maps. I understand that the definition given here rests upon Topology by Munkres, and I do not have it at the moment.--Julius (talk) 09:43, 16 December 2022 (UTC)

By definition of path, a path is always a continuous mapping. By the general definition of loop, a loop is a path, so a loop is a continuous mapping. Arguably in this definition of loop in topological manifolds, the word continuous is superfluous. --Anghel (talk) 12:22, 16 December 2022 (UTC)

Good point. I will check my sources later tonight (probably the author prefers recalling the obvious) and update accordingly.--Julius (talk) 14:22, 16 December 2022 (UTC)

It is then worth following up the suggestion that there should not in fact actually be a separate definition for a loop in a topological manifold. IMO there is only a need to sub-specify the definitions if they differ in any way. In this case there is no such difference. (If we wish to emphasise the fact that the mapping is by default continuous, and I don't think we do, we can perhaps put it in brackets, thus: "(continuous) path".)

Very well. We can ditch loops for topological manifolds. Those who care can easily check that a topological manifold is a Hausdorff space which is a topological space.--Julius (talk) 00:53, 18 December 2022 (UTC)