I've started studying algebraic topology and I've come up with my first example of real category: the h-TOP category. In order to understand it better, I've thought about an example, and I'd like to know whether it's correct or not.
I start considering the objects $\mathbb{R}_*$ (which represents $\mathbb{R} \setminus \{0\}$) and $X = D^2 \sqcup D^2$ (the disjoint union of two unit disks, which I represent as a subspace of the real space: $(\{0\} \times D^2) \cup (\{1\} \times D^2) = \{ \vec{x}\ |\ x_1^2 + x_2^2 + x_3^2 \leq 1,\ x_1 = 0, 1 \}$).
On the one hand, the set of morphisms from $\mathbb{R}_*$ to $X$ is precisely the one containing only $[a]$, $[b]$, $[c]$ and $[d]$, where \begin{equation} a(x) = \left\{ \begin{array}{ll} (0, 0, 0) && x < 0 \\ (1, 0, 0) && x > 0 \end{array} \right. \textrm{,} \end{equation} \begin{equation} b(x) = \left\{ \begin{array}{ll} (1, 0, 0) && x < 0 \\ (0, 0, 0) && x > 0 \end{array} \right. \textrm{,} \end{equation} \begin{equation} c(x) = (0, 0, 0) \textrm{,} \end{equation} \begin{equation} d(x) = (1, 0, 0) \textrm{.} \end{equation}
On the other hand, the set of morphisms from $X$ to $\mathbb{R}_*$ is precisely the one containing only $[e]$, $[f]$, $[g]$ and $[h]$, where \begin{equation} e(\vec{x}) = \left\{ \begin{array}{ll} -1 && x_1 = 0 \\ 1 && x_1 = 1 \end{array} \right. \textrm{,} \end{equation} \begin{equation} f(\vec{x}) = \left\{ \begin{array}{ll} 1 && x_1 = 0 \\ -1 && x_1 = 1 \end{array} \right. \textrm{,} \end{equation} \begin{equation} g(\vec{x}) = -1 \textrm{,} \end{equation} \begin{equation} h(\vec{x}) = 1 \textrm{.} \end{equation}
Notice that $[e] \circ [a] = \textrm{Id}_{\mathbb{R}_*}$, because \begin{equation} e \circ a = e\left( \left\{ \begin{array}{ll} (0, 0, 0) && x < 0 \\ (1, 0, 0) && x > 0 \end{array} \right. \right) = \left\{ \begin{array}{ll} -1 && x < 0 \\ 1 && x > 0 \end{array} \right. \end{equation} is homotopic to $\textrm{Id}_{\mathbb{R}_*}$.
Also, $[a] \circ [e] = \textrm{Id}_X$, because \begin{equation} a \circ e = a\left( \left\{ \begin{array}{ll} -1 && x_1 = 0 \\ 1 && x_1 = 1 \end{array} \right. \right) = \left\{ \begin{array}{ll} (0, 0, 0) && x_1 = 0 \\ (1, 0, 0) && x_1 = 1 \end{array} \right. \end{equation} is homotopic to $\textrm{Id}_X$.
From these last observations follows the fact that $[a]$ is an isomorphism between $\mathbb{R}_*$ and $X$ and hence that $\mathbb{R}_*$ and $X$ are homotopically equivalent.
Please, point out EVERY mistake, and suggest other examples that allow me to understand the category h-TOP better.