Combinatorial topology is a subset of algebraic topology that uses combinatorial methods. For example, simplicial homology is a combinatorial construction in algebraic topology, so it belongs to combinatorial topology.

Algebraic topology originated with combinatorial topology, but went beyond it probably for the first time in the 1930s when Čech cohomology was developed.

Alexandrov, P. S. Combinatorial Topology. New York: Dover, 1998.

Pontryagin, L. S. Foundations of Combinatorial Topology. New York: Dover, 1999.

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Weisstein, Eric W. "Combinatorial Topology." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/CombinatorialTopology.html