  Topological deep learning is a formalism that is aimed at introducing
topological language to deep learning for the purpose of utilizing the minimal
mathematical structures to formalize problems that arise in a generic deep
learning problem. This is the first of a sequence of articles with the purpose
of introducing and studying this formalism. In this article, we define and
study the classification problem in machine learning in a topological setting.
Using this topological framework, we show when the classification problem is
possible or not possible in the context of neural networks. Finally, we
demonstrate how our topological setting immediately illuminates aspects of this
problem that are not as readily apparent using traditional tools.
