Euler characteristic

Alternating cell or homology-rank counts define a topological invariant in suitable finite settings.

Different ways to understand it

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Homology groups
Cycles modulo boundaries measure holes through a chain complex.
Simplicial complexes
Simplices fitted together along faces provide combinatorial models of spaces.
Chain complexes
Successive boundary maps compose to zero in a graded sequence of groups or modules.
Cohomology
Dual cochain constructions yield invariants and additional operations such as cup products.
Covering spaces
A covering locally maps disjoint sheets homeomorphically onto an open neighborhood.