A covering space for a topological space X is a topological space C together with a continuous map (see continuous function) π‎:C → X, called the covering map, such that about each point p ∈ X is an open set U such that π‎⁻¹(U) is a disjoint union of open sets in C which are homeomorphically mapped onto U by π‎. For example, the real line ℝ is a covering space for the unit circle S¹ with π‎:ℝ → S¹ given by π‎(x)=(cosx,sinx). The covering map winds ℝ repeatedly onto S¹. The pre-image of the point p =  (1,0) is the set 2π‎ℤ and so a small arc U around (1,0) exists which has pre-image made up of small disjoint intervals (2nπ‎−ε‎, 2nπ‎ + ε‎).

A useful property of covering spaces is that if γ‎ is a path in X there is a unique path Γ‎ in C such that γ‎ = π‎∘Γ‎. The path Γ‎ is called the lift of γ‎.

A universal covering space is a covering space which is simply connected. So in the example above, ℝ is a universal covering space for S¹. A universal covering space of X is a covering space for any connected covering space of X. If a universal covering space exists, it is unique up to homeomorphism.