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

*V*_{a} 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}(*b*_{0}) has *k* elements for some *b*_{0} 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.