This is the first in a series of posts merging ideas from topology with current techniques of machine learning (such as deep generative models). Here we give an introduction to topology and discuss why it is a useful framework for data science. Subsequent posts will go into more technical detail and describe various applications. What is topology? Topology is the mathematical discipline that studies shape, with a fairly lenient notion of what it means for two shapes to be the “same.” To get a feeling for topology, it is useful to contrast it with its more familiar cousin, geometry. In geometry, two shapes are considered the same up to rigid motion: picking up a triangle and rotating or moving it to a different location does not change its shape. ...