The Klein bottle which has no distinction between inside and outside surfaces can be made by gluing two of what similar objects?
Manifold pair of circles, but (topologically) a single circle; and what was once its "inside" has merged with its "outside", so that it now has only a "single" side. Take two Möbius strips; each has a single loop as a boundary. Straighten out those loops into circles, and let the strips distort into cross-caps. Gluing the circles together will produce a new, closed manifold without boundary, the Klein bottle. Closing the surface does nothing to improve the lack of orientability, it merely removes the boundary. Thus, the Klein bottle is a closed surface with no distinction between inside and outside. Note
Solid Klein bottle boundary is a Klein bottle. Solid Klein bottle In mathematics, a solid Klein bottle is a three-dimensional topological space (a 3-manifold) whose boundary is the Klein bottle. It is homeomorphic to the quotient space obtained by gluing the top disk of a cylinder formula_1 to the bottom disk by a reflection across a diameter of the disk. Alternatively, one can visualize the solid Klein bottle as the trivial product formula_2, of the möbius strip and an interval formula_3. In this model one can see that the core central curve at 1/2 has a regular neighborhood which is again a trivial