Algebraic Topology pp 35-81 | Cite as

# The Fundamental Group

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## Abstract

Let *X* be a topological space. Often we associate with *X* an object that depends on *X* as well as on a point *x* of *X.* The point *x* is called a **base point** and the pair (*X, x*) is called a **pointed space**. If (*X, x*) and (*Y, y*) are two pointed spaces, then a continuous map *f : X → Y* such that *f*(*x*) = *y* is called a map between pointed spaces. Let *f : X → Y* be a homeomorphism and *x* be a point of *X.* Then *f* is a homeomorphism between pointed spaces (*X, x*) and (*Y, f*(*x*)). The composite of two maps between pointed spaces is again a map between pointed spaces and the identity map *I*_{(X,x)} : (*X, x*) *→* (*X, x*) is always a homeomorphism of pointed spaces for each *x* ∈ *X*.

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