# A Numerical Approach for the Filtered Generalized Cech Complex
## 3 Filtered Generalized Simplicial Structures for Disk Systems  
In this section, we introduce two simplicial structures associated with a disk system \(M\), as well as the filtration induced by rescaling the system \(M\). The importance of these notions lies in their relation to the topological analysis through persistent homology of filtered simplicial structures, induced by point clouds with nonhomogeneous neighborhoods. Let \(M\) be a disk system. Denote by \(\mathcal{VR}(M)\) the family of all Vietoris-Rips subsystems of \(M\), this is,  
\[\mathcal{VR}(M)=\{\sigma\subset M\mid D_{i}\cap D_{j}\
eq\emptyset,D_{i},D_{j }\in\sigma\}.\]  
Analogously, denote by \(\mathcal{C}(M)\) the set of all Cech subsystems,  
\[\mathcal{C}(M)=\{\sigma\subset M\mid\cap_{D_{i}\in\sigma}D_{i}\
eq\emptyset\}.\]  
On the other hand, recall that a simplicial structure on a (finite) set \(V\) is defined as a family \(\Delta(V)\subset 2^{V}\) of subsets of \(V\) such that if \(\sigma\in\Delta(V)\) and \(\tau\subset\sigma\), then \(\tau\in\Delta(V)\). Thus, for any \(\sigma\in\mathcal{VR}(M)\) every disk subsystem \(\tau\subset\sigma\) is also in the family \(\tau\in\mathcal{VR}(M)\). The same property is valid for the family \(\mathcal{C}(M)\). These properties imply that \(\mathcal{VR}(M)\) and \(\mathcal{C}(M)\) are simplicial complexes. We refer to \(\mathcal{VR}(M)\) as the generalized Vietoris-Rips complex associated to the disk system \(M\), and to \(\mathcal{C}(M)\) as the generalized Cech complex of \(M\). The above construction allows us to perform topological data analysis of point cloud data through the persistent homology of the generalized Vietoris-Rips or Cech complexes. However, to perform such analysis it is necessary to construct a filtered simplicial structure. We will define a filtration through weight functions. Let \(\Delta\) be a simplicial complex and let \(\omega:\Delta\to\mathbb{R}\) be a function. We call \(\omega\) a weight function over the simplicial complex \(\Delta\) if \(\tau,\sigma\in\Delta\) and \(\tau\subset\sigma\), implies \(\omega(\tau)\leq\omega(\sigma)\). For example, to the generalized Cech complex \(\mathcal{C}(M)\) of the disk system \(M\), the function \(\omega:\mathcal{C}(M)\to\mathbb{R},\sigma\mapsto\mu_{\sigma}\), which assigns the Cech scale to any Cech subsystem \(\sigma\subset M\), is a weight function, called the Cech-weight function. The analogous property holds for the Vietoris-Rips complex and the Vietoris-Rips scale (see [18] for the construction of the filtered generalized Cech complex using weighted point clouds). Moreover, from the definition we have for the Cech-weight function and to every non-negative scale \(\lambda\geq 0\), that  
\[\omega^{-1}((-\infty,\lambda])=\mathcal{C}(M_{\lambda}).\]  
We denote by \(\mathcal{C}_{M}(\lambda)\) the family \(\mathcal{C}(M_{\lambda})\) for \(\lambda\geq 0\), i.e., the family of all Cech subsystems of the \(\lambda\)-rescaled disk system \(M_{\lambda}\), in order to make the dependence explicit with respect the the parameter. We establish the analogous definition for \(\mathcal{VR}_{M}(\lambda)\), for any \(\lambda\geq 0\). Note that there is no restriction on the scale \(\lambda\geq 0\), additional to the non-negativity, i.e., we allow greater values of \(\lambda\) than \(1\), in the interest of studying the generalized Cech complex of rescaled disk systems beyond the original. For \(\lambda^{\prime}\leq\lambda\) we have the families contention: \(\mathcal{VR}_{M}(\lambda^{\prime})\subset\mathcal{VR}_{M}(\lambda)\) and \(\mathcal{C}_{M}(\lambda^{\prime})\subset\mathcal{C}_{M}(\lambda)\). In general, given a simplicial complex \(\Delta\) and a weight function \(\omega:\Delta\to\mathbb{R}\), any increasing sequence \(\lambda_{1}<\cdots<\lambda_{s}\) of real numbers induces a simplicial filtration: \(\Delta_{1}\subset\cdots\subset\Delta_{s}\) for \(\Delta_{i}:=\omega^{-1}((-\infty,\lambda_{i}])\).