Let $Y$ be a Hausdorff space, and let $f:S^{n-1} \to Y$ be continuous. Then $D^n \coprod_f Y$ is called the space obtained from $Y$ by attaching an n-cell (denoted $e^n$) via $f$ and is denoted $Y_f$
The charactersitc map $\Phi$ is the composite:
$D^n \hookrightarrow D^n \coprod Y \to D^n \coprod_f Y \to Y_f$ so that $\Phi:(D^n,S^{n-1}) \to (Y_f,Y)$ is a function of pairs, and $\Phi|S^{n-1}$ is the attaching map.
(See here for $D^n \coprod_f Y$ if this is non-standard).
As a (presumably) gentle introduction to CW-complexes I have the following question:
If $Y$ is a singleton, show that the space obtained from $Y$ by attaching an n-cell is $S^n$, hence $S^n = e^0 \cup e^n$ (where $e^n \simeq D^n - S^{n-1}$).
I am unsure how to approach this problem. If I let $y=(-1,0,\ldots,0)$, then I believe there is a map $\Phi:(D^n,S^{n-1}) \to (S^n, y)$, such that $\Phi|e^n$ is an embedding.