I’ve been watching tutorials in modeling and have come across terms like topology and manifold. These terms are common in mathematics related to studies on calculus, so I was wondering how similar are these concepts to the ones in mathematics, if at all? Thanks.
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Those terms are supposed to refer to same mathematical notions.
Topology is relative to the wireframes of meshes in 3D modeling, not to surfaces.
In a polygonal modeler like blender, your 3D model is just a surface.
So, you can create a manifold or a non-manifold 3D model that will mathematically correspond to manifold or non-manifold surface.
By simplification, selecting non-manifold elements of a mesh means selecting mesh elements at a boundary or a self-intersection of this surface.
Actually I still find the different notions confusing, also.
In 3d topology is about mesh connectivity, not about a notion of continuous functions in relation to the surface.
But if you do not use the calculus definition of topology but the more generic set-theoretic definition, the mesh connectivity can be described as a topology on the set of vertices/edges/faces.