TITLE: How to see the real projective plane is a Möbius band glued to a disk?
QUESTION [15 upvotes]: I am seeking an easily comprehended, convincing explanation 
that ${RP}^2$ is topologically the same as gluing the circle
boundary of a disk to the edge of a Möbius band.

REPLY [2 votes]: It's really the same thing as Daniel's answer, but I find this easier to visualize:
Consider $\mathbb{RP}^2$ as a quotient of $S^2$. The image of the disk $\{(x,y,z)\in S^2\mid z\geq1/2\}$ is the disk, and the image of the strip $\{(x,y,z)\in S^2\mid y\geq0,\,|z|\leq1/2\}$ is the Möbius strip.