Define what is meant by a Möbius band. Identify the space obtained by identifying the boundary of a Möbius band to a point. Give a brief explanation.
Solution
1. A Möbius band is the quotient\[ M=([0,1]\times[-1,1])/{\sim}, \qquad (0,t)\sim(1,-t). \]
This is the standard rectangle model: the two vertical edges are glued with a half-twist.
2. The boundary \(\partial M\) is a single circle.
The boundary is the image of \[ [0,1]\times\{-1,1\}. \] Under the twisted edge identification, \[ (0,1)\sim(1,-1), \qquad (0,-1)\sim(1,1), \] so the two horizontal edges join end-to-end to form one circle.
3. In the quotient \(M/\partial M\), a collar of \(\partial M\) becomes a disk.
Choose a closed collar \[ N\cong S^1\times[0,1] \] of the boundary, with \[ \partial M\cong S^1\times\{0\}. \] After collapsing \(\partial M\) to one point, the image of \(N\) is \[ (S^1\times[0,1])/(S^1\times\{0\}), \] the cone on \(S^1\). The cone on \(S^1\) is homeomorphic to the closed disk \(D^2\) via \[ [(e^{i\theta},r)]\longmapsto re^{i\theta}. \] Its boundary corresponds to \(S^1\times\{1\}\).
4. The quotient \(M/\partial M\) is obtained by attaching a disk to a Möbius band along its boundary.
Remove the interior of the collar \(N\) from \(M\). The remaining surface \[ M'=M\setminus\operatorname{int}(N) \] is again a Möbius band, with boundary \(S^1\times\{1\}\). By 3, the collapsed collar becomes a disk whose boundary is exactly this circle. Thus \[ M/\partial M\cong M'\cup_{\partial M'}D^2. \]
5. Therefore\[ \boxed{M/\partial M\cong\mathbb{RP}^2}. \]
The real projective plane is obtained by attaching a \(2\)-disk to a Möbius band along its boundary. Indeed, in the disk model \[ \mathbb{RP}^2=D^2/(x\sim -x\text{ on }\partial D^2), \] an annular neighborhood of the boundary descends to a Möbius band, while the remaining central disk caps its boundary. By 4, \(M/\partial M\) has exactly this decomposition.