The Fundamental Group
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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