Open subsets under class-metric

In this section $\{\mathcal{X}^{(\ast)},\rho_b\}$ is the metric space of patches with triangles generated with $acl(2,1)_{110}$ . Let be $x_0\in\mathcal{X}^{(\ast)}$ and $k\in\mathbb{Z^+}$ . A open ball, named by $S(x_0,k)$ , is defined as the set of patches

$$S(x_0,k)\leftrightharpoons\{x\in\mathcal{X}^{(\ast)}:\rho_b(x_0,x)<k\}$$

The center of the open ball is $k$ , for simplicity, we call it ``ball''. The elements of the metrical space $\{\mathcal{X}^{(\ast)};\rho_b\}$ can be classified in some categories:

• Let $Y\subset\mathcal{X}^{(\ast)}$ a subset of patches. A patch $x\in\mathcal{X}^{(\ast)}$ is a contact element to set $Y$ , if any open ball with center in $x$ contents at least one patch in $Y$ . The set of all contact elements of $Y$ is called the closure of $Y$ and we'll write it $\overline{Y}$

• Let $Y\subset\mathcal{X}^{(\ast)}$ and $x\in\mathcal{X}^{(\ast)}$ a contact patch. $x$ is an isolated patch if the ball with center in $x$ contents no other patches of $Y$ than $x$ .

• Let $Y\subset\mathcal{X}^{(\ast)}$ and $\widetilde{Y}$ its complement. A patch of $x\in\mathcal{X}^{(\ast)}$ is an interior patch of the set $Y$ if exist a ball $S(x,r)$ such that $S(x,r)\subset Y$ . The set of all interior patches of $Y$ is called the interior of $Y$ and we'll write it $\overset{\circ}{Y}$

• $Y\subset\mathcal{X}^{(\ast)}$ is an open subset of $\mathcal{X}^{(\ast)}$ if any patch that is an element of $Y$ is an isolated patch, that is $Y=\overset{\circ}{Y}$ .

• A patches subset $Z\subset\mathcal{X}^{(\ast)}$ is a closed subset of $\mathcal{X}^{(\ast)}$ if $Z=\overline{Z}$

The above definitions have sense when we need to define open sets, particularly in the next section 2.2, where we will define tiles, useful for cover the 2-space. The metric space $\{\mathcal{X}^{(\ast)};\rho\}$ also defines a topological space with $\mathcal{X}^{(\ast)}$ and $\mathbb{P}(\mathcal{X}^{(\ast)})$ as the set of all subsets of $\mathbb{P}(\mathcal{X}^{(\ast)})$ . Thus the $(\mathcal{X}^{(\ast)},\mathbb{P}(\mathcal{X}^{(\ast)}))$ system is the topological space of all patches generated with $lca(2,1)_{110}$ .