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### Introduction

Topology is concerned with the properties of a geometric shape that are unchanged when we continuously deform it. In more technical terms, topology tries to classify TOPOLOGICAL SPACES [III.90], where two spaces are considered the same if they are homeomorphic. Algebraic topology assigns numbers to a topological space, which can be thought of as the “number of holes” in that space. These holes can be used to show that two spaces are not homeomorphic: if they have different numbers of holes of some kind, then one cannot be a continuous deformation of the other. In the happiest cases, we can hope to show the converse statement: that two spaces with the same number of holes (in some precise sense) ...

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