I've been given the following definition of a mapping cone: We have a two topological spaces $X,Y$ and a continuous map $f: X' \rightarrow Y$ where $X'$ is a subspace of $X$. Then we define an equivalence relation $\sim$ on $Y \sqcup X$ as follows: for any $x_1, x_2 \in X'$ such that $f(x_1)=f(x_2)=y \in Y$ we set $x_1 \sim x_2 \sim y$. The Quotient Space is denoted as $Y \cup_fX$
The Mapping Cylinder $M_f$ of a continuous $f:X\rightarrow Y$ is defined as $Y \cup_{f_0} (X \times [0,1])$ where $f_0:X \times \{1\} \rightarrow Y$ is given by $f_0(x,1)=f(x)$.
And the Mapping Cone is $C_f=M_f/(X \times \{0\})$
I now have to show that the mapping cone of the natural map $S^n \rightarrow \mathbb{RP}^n$ is homeomorphic to $\mathbb{RP}^{n+1}$
But I'm just having way too much trouble with translating the given definition to figure out the problem. Any kind of assistance is appreciated!