$\mathbb{R}\setminus F$ is homotopy equivalent to the discrete set with $n+1$ points Let $F$ be a set of $n$ real numbers. Show that $\mathbb{R}\setminus F$ is homotopy equivalent to the discrete space with $n+1$ points.
Hence give a calculation of $\pi_0(\mathbb{R}\setminus F)$, where $\pi_0(X)$ are the path components of a space $X$.
I'm having  little trouble getting started here. I have drawn a picture and can clearly see what the question is asking for, but I don't know how to begin formally writing this down
 A: Since each open interval is homotopy equivalent to a single point, the gluing lemma implies that $\mathbb R\setminus F$ is homotopy equivalent to the discrete space with $n+1$ points, as $\mathbb R\setminus F$ is the union of $n+1$ disjoint open intervals. More precisely, if we write $X_1,\cdots,X_{n+1}$ for the $n+1$ disjoint open intervals and arbitrarily pick $x_i\in X_i$, then 
$$
p_i\circ\iota_i\simeq\mathrm{id}_{\{x_i\}},\iota_i\circ p_i\simeq \mathrm{id}_{X_i},
$$
where $p_i\colon X_i\to \{x_i\}$ and $\iota_i\colon\{x_i\}\to X_i$ are respectively the projection to $x_i$ and the inclusion map. Let us write $H_i\colon\{x_i\}\times I\to\{x_i\}$ and $F_i\colon X_i\times I\to X_i$ for these two homotopies ($I=[0,1]$). Then the gluing lemma implies that
$$
H\colon \{x_1,\cdots,x_{n+1}\}\times I\to \{x_1,\cdots,x_{n+1}\}
$$ and $F\colon\mathbb R\setminus F\times I\to\mathbb R\setminus F$, which are defined by $H|_{\{x_i\}\times I}=H_i$ and $F|_{X_i\times I}=F_i$, are continuous. Thus
$$
p\circ\iota\overset{H}{\simeq}\mathrm{id}_{\{x_1,\cdots,x_{n+1}\}},\iota\circ p\overset{F}{\simeq}\mathrm{id}_{\mathbb R\setminus F},
$$
where $p\colon\mathbb R\setminus F\to \{x_1,\cdots,x_{n+1}\}$ is defined by $p|_{X_i}=p_i$ and $\iota\colon\{x_1,\cdots,x_{n+1}\}\to\mathbb R\setminus F$ is defined by$\iota|_{\{x_i\}}=\iota_i$ (Their continuity is also obtained by the gluing lemma). Hence $\mathbb R\setminus F$ is homotopy equivalent to $\{x_1,\cdots,x_{n+1}\}$.
The path components of $\mathbb R\setminus F$ are exactly these $n+1$ disjoint open intervals.
