  We utilize classical facts from topology to show that the classification
problem in machine learning is always solvable under very mild conditions.
Furthermore, we show that a softmax classification network acts on an input
topological space by a finite sequence of topological moves to achieve the
classification task. Moreover, given a training dataset, we show how
topological formalism can be used to suggest the appropriate architectural
choices for neural networks designed to be trained as classifiers on the data.
Finally, we show how the architecture of a neural network cannot be chosen
independently from the shape of the underlying data. To demonstrate these
results, we provide example datasets and show how they are acted upon by neural
nets from this topological perspective.
