Let p \colon \tilde{X} \to X be a continuous map. An open set U \subset X is evenly covered if p^{-1}(U) is a union of disjoint open sets of \tilde{X}, each of which is mapped homeomorphically onto U by p. The map p is a covering map, and \tilde{X} is a covering space of X, when every point of X has an evenly covered open neighborhood [1,2]. The disjoint open sets over U are called the sheets over U.
Each sheet over an evenly covered U meets the fiber p^{-1}(x) in exactly one point for every x \in U, so the number of points in a fiber does not change inside U. The number of points of p^{-1}(x) is therefore a locally constant function of x, and when X is connected it is a constant, the number of sheets of the covering. It may be infinite, and a covering with n sheets is called an n-sheeted covering.
The map p \colon \mathbb{R} \to S^1, p(s) = (\cos 2\pi s, \sin 2\pi s), is a covering with one sheet for each integer, since the preimage of an open arc shorter than the circle is a disjoint union of open intervals. With S^1 \subset \mathbb{C}, the map z \mapsto z^n is an n-sheeted covering of S^1 by itself. The quotient S^n \to \mathbb{RP}^n identifying antipodal points is 2-sheeted, as an open set in an open hemisphere misses its antipodal image.
The figure cuts the circle at the base point x_0 and draws the 3-sheeted covering \mathbb{R}/3\mathbb{Z} \to S^1, s \mapsto (\cos 2\pi s, \sin 2\pi s), as three copies of the cut circle stacked above it. The numbers at the ends are values of s, and equal numbers are glued. When the arc U contains x_0, each of its three preimages is split between the end of one copy and the start of the next, yet it is still one arc mapped homeomorphically onto U.
A surjective local homeomorphism need not be a covering map. The same formula on the open interval (0, 3) is one, but the point x_0 has only the two preimages s = 1 and s = 2, while every other point has three. In the second half of the figure, an arc around x_0 has two pieces of its preimage near the missing ends s = 0 and s = 3, and each of them is mapped onto only half of the arc. No neighborhood of x_0 is evenly covered.
A covering map has the path lifting property: for every path \gamma in X starting at x_0 and every \tilde{x} \in p^{-1}(x_0), there is exactly one path \tilde{\gamma} starting at \tilde{x} with p \circ \tilde{\gamma} = \gamma. Homotopies lift in the same way, so paths that are homotopic with fixed endpoints have lifts from \tilde{x} with the same endpoint [1]. These tools gave the fundamental group of the circle, and they work for any covering.
For \tilde{x} \in p^{-1}(x_0) and a loop \gamma at x_0, let \tilde{x} \cdot [\gamma] be the endpoint of the lift of \gamma from \tilde{x}. Lifting \gamma and then \delta gives (\tilde{x} \cdot [\gamma]) \cdot [\delta] = \tilde{x} \cdot [\gamma\delta], so \pi_1(X, x_0) acts on the fiber from the right. If \tilde{X} is path-connected, a path from \tilde{x} to \tilde{y} projects to a loop moving \tilde{x} to \tilde{y}, and the action is transitive.
The homomorphism p_* \colon \pi_1(\tilde{X}, \tilde{x}_0) \to \pi_1(X, x_0) is injective: if p \circ \tilde{\gamma} is null-homotopic, lifting the null-homotopy starting at \tilde{x}_0 gives a null-homotopy of \tilde{\gamma}. Its image consists exactly of the classes of loops whose lifts starting at \tilde{x}_0 are again loops [1]. In terms of the action, H = p_*\pi_1(\tilde{X}, \tilde{x}_0) is the set of classes that fix \tilde{x}_0.
Two classes [\gamma] and [\delta] move \tilde{x}_0 to the same point exactly when [\gamma][\delta]^{-1} fixes \tilde{x}_0, that is, when they lie in the same right coset of H. When \tilde{X} is path-connected, H[\gamma] \mapsto \tilde{x}_0 \cdot [\gamma] is therefore a bijection from the right cosets of H to the fiber, and if X is also path-connected, the number of sheets equals the index of H in \pi_1(X, x_0) [1].
For z \mapsto z^n on the circle, H is n\mathbb{Z} \subset \mathbb{Z}, of index n. The torus S^1 \times S^1 has the 2-sheeted covering (z, w) \mapsto (z^2, w), and its H is 2\mathbb{Z} \times \mathbb{Z}: a loop lifts to a loop exactly when it goes around the first circle an even number of times. For S^2 \to \mathbb{RP}^2 the group H is trivial, so the nontrivial element of \pi_1(\mathbb{RP}^2) \cong \mathbb{Z}/2\mathbb{Z} exchanges the two points of the fiber.
The figure shows a 3-sheeted covering of the figure eight S^1 \vee S^1 as a graph whose vertices 1, 2, 3 form the fiber. Edges labeled a cover the left circle and edges labeled b the right one; a exchanges 1 and 2 and fixes 3, and b exchanges 2 and 3 and fixes 1. The lifts of b, a, ab and a^2 from the vertex 1 end at 1, 2, 3 and 1. So b and a^2 belong to H, and H, Ha and Hab are the three cosets, one for each sheet.
Paths are not the only maps that lift. Let p \colon (\tilde{X}, \tilde{x}_0) \to (X, x_0) be a covering map and f \colon (Y, y_0) \to (X, x_0) a map from a path-connected and locally path-connected space Y. Then there is a lift \tilde{f} \colon (Y, y_0) \to (\tilde{X}, \tilde{x}_0), a map with p \circ \tilde{f} = f, if and only if f_*\pi_1(Y, y_0) \subset p_*\pi_1(\tilde{X}, \tilde{x}_0) [1].
The condition is necessary since f_* = p_* \circ \tilde{f}_*. Conversely, let \tilde{f}(y) be the endpoint of the lift from \tilde{x}_0 of f \circ \eta, where \eta is a path from y_0 to y. For two such paths \eta_1 and \eta_2, the class of f \circ (\eta_1 \bar{\eta}_2) lies in p_*\pi_1(\tilde{X}, \tilde{x}_0), so it lifts to a loop and both give the same point. Local path-connectedness, applied inside evenly covered sets, makes \tilde{f} continuous.
Once one value is fixed, the lift is unique: if Y is connected and two lifts of f agree at one point, they agree everywhere [1]. Over an evenly covered neighborhood of f(y), two lifts near y either lie in the same sheet or in different sheets, so the set where they agree and the set where they differ are both open.
Every map to S^1 from a simply connected, locally path-connected space lifts to \mathbb{R}, since the condition f_*\pi_1(Y) \subset p_*\pi_1(\mathbb{R}) holds when the left side is trivial. A map f \colon S^2 \to S^1 thus factors as p \circ \tilde{f} with \tilde{f} \colon S^2 \to \mathbb{R}. As \mathbb{R} is contractible, \tilde{f} and hence f are null-homotopic, and the same holds for maps S^n \to S^1 with n \geq 2.
Local path-connectedness cannot be dropped. The Warsaw circle W in the figure is the graph of \sin(\pi/x) for 0 < x \leq 1, together with the segment from (0, -1) to (0, 1) on which the graph accumulates, and an arc below joining the two. No path enters the segment from the graph, so every loop in W is null-homotopic and \pi_1(W) is trivial [1]. Let f \colon W \to S^1 send each point to its direction as seen from the point c in the region enclosed by W.
Suppose \tilde{f} were a lift of f, measured in full turns from its value at the bottom of the segment. On the segment its values lie between -0.16 and 0. Around the arc and along the graph toward the segment, the angle grows by almost a full turn, and near the segment the value stays above 0.8. The graph comes arbitrarily close to every point of the segment, so \tilde{f} cannot be continuous, although f_*\pi_1(W) is trivial.
A covering p \colon \tilde{X} \to X with \tilde{X} simply connected is called a universal cover. If X is locally path-connected and q \colon (Y, y_0) \to (X, x_0) is another covering with Y path-connected, the lifting criterion applies to p, because p_*\pi_1(\tilde{X}) is trivial and lies in q_*\pi_1(Y, y_0). So p factors through every connected covering of X, which is the reason for the name [1,2].
A universal cover exists when X is path-connected, locally path-connected and semilocally simply connected, which means that every point has a neighborhood U in which every loop is null-homotopic in X. The last condition is also necessary. A loop in an evenly covered U lifts into a single sheet, the lift is null-homotopic because the covering space is simply connected, and projecting the null-homotopy shows that the loop is null-homotopic in X.
The construction takes as points of \tilde{X} the classes [\gamma] of paths in X starting at x_0, up to homotopy fixing both endpoints, and sets p([\gamma]) = \gamma(1). A basic neighborhood of [\gamma] consists of the classes [\gamma\eta] with \eta a path in a small neighborhood of \gamma(1) as above. The fiber over x_0 is the set of classes of loops, which is \pi_1(X, x_0), and the group acts on it by multiplication from the right [1,2].
The coverings \mathbb{R} \to S^1, \mathbb{R}^2 \to S^1 \times S^1 and S^n \to \mathbb{RP}^n with n \geq 2 are universal. The universal cover of the figure eight is the tree with four edges at every vertex, two labeled a and two labeled b, whose vertices correspond to the reduced words in a, b and their inverses. The Hawaiian earring, the union of the circles of radius 1/n centered at (1/n, 0), is not semilocally simply connected at the origin and has no universal cover.
The figure shows S^1 \vee S^2, a circle and a sphere joined at x_0. Its universal cover is the line \mathbb{R} with a copy of S^2 attached at each integer, and the integers form the fiber over x_0. A loop around the sphere lifts to a loop on one sphere, and a loop around the circle lifts to a path to the next integer, matching \pi_1(S^1 \vee S^2, x_0) \cong \mathbb{Z}. Shrinking the line to a point shows that this cover is homotopy equivalent to a wedge of infinitely many 2-spheres.