Figure 1

Example of persistent homology for a point cloud. (a) A finite set of points in \(\mathbb{R}^{2}\) (for \(\epsilon=0\)) and a nested sequence of spaces obtained from it (from \(\epsilon=0\) to \(\epsilon=2.1\)). (b) Barcode for the nested sequence of spaces illustrated in (a). Solid lines represent the lifetime of components, and dashed lines represent the lifetime of holes.