Homotopy theory is a branch of algebraic topology that studies spaces and their continuous transformations, focusing on the idea that spaces can be deformed into one another. It captures the notion of 'shape' in a very abstract way, allowing mathematicians to classify spaces based on their homotopy equivalence rather than their rigid geometric structure. This perspective is essential in understanding how to analyze data in topological data analysis, where shapes and spaces represent the underlying structure of data.
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