Let $M$ be the mobius strip which we define as quotient space of $X:=[-1,1]\times [0,1]$ with the equivalence relation $(t,0)\sim(-t,1)$, i.e. $M=X/\sim$.
Let $\partial M$ be the boundary of $M$, i.e. the subspace consisting of elements with representatives of the form $(t,s)$ for $t\in\{-1,1\}$ and $s\in[0,1]$.
Let $i:\partial M\rightarrow M$ such that $i(x)=x$.
My task is to construct a pushout of the following form
here $\mathbb{R}\mathbb{P}^2=\{\text{span}(x):x\in\mathbb{R}^3\setminus\{0\}\}$ denotes the 2 dimensional projective space.
i.e. it consists of all 1 dimensional subspaces of $\mathbb{R}^3\setminus\{0\}$.
$D^2$ denotes the 2 dimensional unit ball, i think, but i am not certain, because it is not stated in the task.
i have no idea how to proceed.
could someone give me a hint please?
i am not sure how to which space should map.
and then the maps from $D^2$ and $M$ have to commute, but i do not see what maps i could choose.
One thing that helps is to play around with cutting and pasting to obtain alternate quotient descriptions for $M$ and for $\mathbb{RP}^2$. The maps in your pushout diagram will be compositions of various homeomorphisms between different cut/paste quotient descriptions.
For example, the Möbius strip is homeomorphic to the quotient of the annulus $$A = \{p \in \mathbb{R}^2 \mid 1 \le |p| \le 2\} $$ defined by the equivalence equivalence relation $p \sim -p$ for each $p \in A$ such that $|p|=2$. You can play around with cutting and pasting to convince yourself that this $A$-quotient and the $X$-quotient in your question are homeomorphic.
For another example, $\mathbb{RP}^2$ is homeomorphic to the quotient of the unit 2-sphere $S^2 \subset \mathbb{R}^3$ defined by equivalence relation $q \sim -q$ for each $q \in S^2$ (lines through the origin of $\mathbb{R}^3$ correspond one-to-one to antipodal point sets $\{p,-p\}$ in $S^2$).
Closely related to this is another description of $\mathbb{RP}^2$ as being homeomorphic to the quotient of a disc by identifying opposite points on the boundary of that disc (the disc is homeomorphic to the closed upper hemisphere of $S^2$, and lines through the origin correspond one-to-one with singletons in the interior of the upper hemisphere together with antipodal point sets on the equator). For convenience, I'll do this with the disc of of radius 2, which I'll denote $$2 D^2 \subset \{p \in \mathbb{R}^2 \mid |p| \le 2\} $$ Then one uses the equivalence relation $p \sim -p$ for each $p \in 2D^2$ such that $|p|=2$. Let me use $P : 2D^2 \to \mathbb{RP}^2$ for that quotient map.
Finally, let $D^2 \subset \mathbb{R}^2$ denote the unit-radius closed disc (yes, that's what it means).
Then the composition of inclusion and quotient map $$A \hookrightarrow 2D^2 \xrightarrow{P} \mathbb{RP}^2 $$ induces a map $$M \mapsto \mathbb{RP}^2 $$ and the composition of the composition of inclusion and quotient map $$D^2 \hookrightarrow 2D^2 \xrightarrow{P} \mathbb{RP}^2 $$ induces a map $D^2 \to \mathbb{RP}^2$. The two maps $M \mapsto \mathbb{RP}^2$ and $D^2 \mapsto \mathbb{RP}^2$ give the pushout diagram you are asking for.