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This means the output neuron in the ReLU network is making decisions based on a strong, well-separated signal, while the Sigmoid network is forced to classify using a weak, compressed one. The key takeaway is that ReLU preserves distance from the decision boundary across layers, allowing that information to compound, whereas Sigmoid progressively destroys it.

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Definition 2 A homeomorphism between topological spaces \(X\) and \(Y\) is a bijective map \(f: X \to Y\) such that both \(f\) and its inverse \(f^{-1}: Y \to X\) are continuous. Explicitly:

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