Homemorphism from projective plane to S1 and Moebius strip

Question

Let $h$ be a homemorphism from $S^{1}$ to the border of the Möbius strip $M$. Also, let $X$ be the quotient of the disjoint union of the closed unit disk $D^{2}$ and $M$ by the equivalence relation that identifies every $x$ in $S^{1}$ with $h(x)$. Prove that $X$ is a homeomorph to $\mathbb{R}P^{2}$ (projective plane over $\mathbb{R}$).

I have no idea how to start this one. My previous exercises asked:

1. Prove that $\mathbb{R}P^{2}$ is homeomorph to the quotient of the sphere $S^{2}$ by the equivalence relation that identifies each point $x\in S^{2}$ with $-x$
2. Prove that $\mathbb{R}P^{2}$ is homeomorph to the quotient of the closed unit disk $D^{2} \subset \mathbb{R}^{2}$ by the equivalence relation that identifies each point $x\in S^{1}$ with $-x$, where $S^{1}$ denotes the unit circumference, in other words, the border of the disk $D^{2}$
3. Prove that the border of the Möbius strip is homemorph to $S^{1}$

I've done these 3 by miracle. Things I know about topology:

a) Definition of topological space and basic properties

b) Metric spaces and basic properties

c) Definition of homeomorphism and basic properties

d) Definition of continuity and basic properties

Things I don't know (because I'll do them next year):

i) Group Theory

ii) Multivariable calculus

iii) Projective Geometry

Answer 1

My way of looking at this uses the characterization of $\Bbb R P^2$ as $S^2/\sim$, where $\sim$ is the antipodal relation. I will describe the approach loosely, and leave it to you to make an explicit homeomorphism. Before taking this quotient, decompose $S^2$ into two closed disks about the north and south poles, and an equatorial region homeomorphic to $S^1 \times I$. The sphere is then obtained by gluing the boundary circles of the polar discs to the two circles on the boundary of the equatorial region.

Now examine what happens when we identify antipodal points. The two polar disks are obviously identified. The quotient of the equatorial region under this identification is homeomorphic to the Möbius strip (You possibly need to prove this). The identified disk is glued to the Möbius strip by identifying the corresponding boundaries. This shows that you get precisely the described space $X$.