Notebooks

Here, I will write some basic notes on Topological Data Anlaysis.

Cech Complex

\[ \mathcal{C}_{\varepsilon} = \left\{\sigma \subseteq \mathcal{D} \quad\mid\quad \underset{x \in \sigma}{\bigcap} B_{\varepsilon}(x) \neq \varnothing\right\} ,\quad B_{\varepsilon}(x) =\{y\ | \ d(x,y)<\varepsilon\} \]

Vietoris Rips Complex

\[ \mathcal{R}_{\varepsilon} =\{\sigma \subset \mathcal{D} \quad \mid \quad \|x-y\|\leq \varepsilon, \text{ for all } x,y\in \sigma\} \]