If a surface consists of two disks with a single band joining them, it is homeom

Question

If a surface consists of two disks with a single band joining them, it is homeomorphic to a single disk with no bands attached. Based on such an observation argue that any connected surface which is built by adding bands to a collection of disks can in fact be built starting with only one disk. (This observation is of practical importance: The surfaces that knots bound will initially be constructed from several disks. Calculations of knot invariants coming from surfaces are much easier if the surface is described using only one disk.)

Explanation / Answer

If the disk with two twisted bands attached. This surface is homeomorphic to the same surface with the bands untwisted. This is because there exists a one-to-one and onto map that cuts, untwists and reattached the bands in the first surface so that it looks, in 3-space, like the second surface. This mapping is continuous because points that are close to each other on the first surface map to points that are close to each other on the second surface.

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