Your comment sits in the nice gradient between not seeing at all the obvious relationships between deep learning and topology and thinking that deep learning is applied topology.
That helps to visualize how the activation functions, bias and weights (linear transformations) serve to stretch the high dimensional space so that data go into extremes and become easy to put in a high dimension, low dimensional object (the manifold) where is trivial to classify or separate.
Gaining an intuition about this process will make some deep learning practices so much easy to understand.
See? Everything lives in the manifold.
Now for a great visualization about the Manifold Hypothesis I cannot recommend more this video: https://www.youtube.com/watch?v=pdNYw6qwuNc
That helps to visualize how the activation functions, bias and weights (linear transformations) serve to stretch the high dimensional space so that data go into extremes and become easy to put in a high dimension, low dimensional object (the manifold) where is trivial to classify or separate.
Gaining an intuition about this process will make some deep learning practices so much easy to understand.