Compacity of graphical patterns on $lca(2,1)_{110}$

The general idea is to make sure that tile has not holes; then let $\{\mathcal{X};\rho_b\}$ be the metric space used and defined in last sections; $p\in\mathcal{X}$ ; $\{Y;\rho_b\}$ the metric subspace; and $Y\leftrightharpoons\{c:c\in p\}$ the set of all cell that are in $p$ . To determine the compacity of $Y$ , we will to determine:

1. All covers of $Y$ .
2. For each cover, find a finite open sub-cover.

First, remember that a cover of $Y$ is a set of subsets $\{Y_j;j\in J\}$ such that $Y\subseteq \bigcup_{j\in J} Y_j$ ; where $J$ is any numerable set. So, the cover of $Y$ is any set of cells such that $Y$ is contained into any union of subsets of $Y$ -cells. This generate a infinite number of sets. But for each cover set of them, the $Y$ set is a finite subcover, because $\vert Y\vert<\infty$ and $Y=\bigcup_{j\in J} Y_j$ . Then a finite subcover is $\bigcup_{j\in J} Y_j$ .

The next step is to show that $\{p\}$ is an open set too. Since we are treating with cells in $p$ , we will probe that $p$ define a topological subspace of $(\mathcal{X};\mathbb{P}(\mathcal{X}))$ . This system uses $Y$ as base set with $\mathbb{P}(Y)$ and the basic metric $\rho _b$ . Thes the system $(Y;\mathbb{P}(Y))$ defines a topological space because:

• $\emptyset \in Y$
• $Y_i\cap Y_j\in Y$ , for $Y_i,Y_j\in\mathbb{P}(Y)$
• $Y_i\cup Y_j\in Y$ , for $Y_i,Y_j\in\mathbb{P}(Y)$

Then $Y$ set is open, consequently the $\{Y;\rho_b\}$ metric subspace is open and it is compact.