1  Introduction

“The unexpected elation with which I had talked about mathematics had suddenly evaporated, and I sat beside him, feeling the weight of my own body, its unnecessary size. Outside of mathematics we had nothing to say to each other, and we both knew it. Then it occurred to me that the emotion with which I had spoken of the blessed role of mathematics on the voyage was a deception. I had been deceiving myself with the modesty, the serious heroism of the pilot who occupies himself, in the gaps of the nebulae, with theoretical studies of infinity. Hypocrisy. For what had it been, really? If a castaway, adrift for months at sea, has a thousand times counted the number of wood fibers that make up his raft, in order to keep sane, should he boast about it when he reaches land? That he had the tenacity to survive? And what of it? Who cared? Why should it matter to anyone how I had filled my poor brain those ten years, and why was that more important than how I had filled my stomach?”
— Stanisław Lem, in “Return from the stars”

Topological data analysis is an exotic animal.

How can one mix a field from abstract math (topology) with real world data? In topology, finite topological spaces are trivial and often ignored, but every real-world data is finite. To connect these two areas of study, we need to have tools that transform finite metric spaces into objects that topology can handle and say something interest about.