Algebraic topology

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Known anciently as combinatorial topology, algebraic topology study the algebraic phenomena involved in topological objects.

For example is is evident for some mathematicians that the set of all functions $S^1\to X$ , where $S^1$ is the circle and $X$ any topological space, is a group, but under the equivalence relation of homotopy, is the important concept of the fundamental group attached to the given space $X$ .