Covering spaces

A covering locally maps disjoint sheets homeomorphically onto an open neighborhood.

Different ways to understand it

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Fundamental group
Based loops modulo endpoint-fixed homotopy form a group under concatenation.
Homotopy equivalence
Maps with inverse compositions homotopic to identities describe spaces with the same homotopy type.
Topological spaces
A topology specifies open sets closed under arbitrary unions and finite intersections.
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.