Let
E and
B be two
topological spaces. Let
p :
E →
B be a
continuous surjective map. An
open set U of
B is said to be
evenly covered by
p if the
inverse image p-1(
U) is equal to the
union of
disjoint sets
Va open in
E. If every point
b of
B has a
neighborhood U that is evenly covered by
p, then
p is called a
covering map and
E is called a
covering space of
B.
Covering maps are important in algebraic topology and are used to prove the Unique Path Lifting Theorem and Unique Path Homotopy Lifting Theorem, which are, in turn, used to determine the Fundamental Groups of certain spaces.
If B is connected and p-1(b0) has k elements for some b0 in B, then p-1(b) has k elements for every b in B. Such maps are called k-fold covering maps and E is called a k-fold covering of B.
If B is Hausdorff, regular, completely regular, or locally compact Hausdorff, then so is E.
If B is compact and p-1(b) is finite for every b in B, then E is compact. This shows that the space E locally resembles the product space B × p-1(b). In fact, covering maps are precisely those maps whose fibers are discrete, i.e., the subspace topology of p-1(b) with respect to E is the discrete topology for every b in B.
If the above conditions hold, we may not only conclude that E is compact, but also that p is a perfect map, i.e., p is a continuous, closed, surjective map, whose fibers are compact. Incidentally, all covering maps are also open maps.