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is called a homotopy from the map <math>h_0</math> to the map <math>h_1</math>, where <math>h_t(x) = h(x, t)</math>. Intuitively, we may think of <math>h</math> as a path from the map <math>h_0</math> to the map <math>h_1</math>. Indeed, a homotopy can be shown to be an [[equivalence relation]]. When ''X'', ''Y'' are pointed spaces, the <math>h_t</math> are required to preserve the basepoints.
Given a pointed space ''X'' and an [[integer]] <math>n \ge 0</math>, let <math>\pi_n(X) = [S^n, X]_*</math> be the homotopy classes of based maps <math>S^n \to X</math> from a (pointed) ''n''-sphere <math>S^n</math> to ''X''. As it turns out, for <math>n > 0</math>, <math>\pi_n(X)</math> are [[group (mathematics)|group]]s called [[homotopy group]]s; in particular, <math>\pi_1(X)</math> is called the [[fundamental group]] of ''X''. Every group is the fundamental group of some space.<ref>{{
If one prefers to work with a space instead of a pointed space, there is the notion of a [[fundamental groupoid]] (and higher variants): by definition, the fundamental groupoid of a space ''X'' is the [[category (mathematics)|category]] where the [[object (category theory)|objects]] are the points of ''X'' and the [[morphism]]s are paths.
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