A covering space is a topological space that "covers" another space, called the base space, in a specific way. Each point in the base space has a neighborhood that is evenly covered by the covering space, meaning that the preimage of that neighborhood consists of disjoint open sets. This concept is important in algebraic topology, as it helps in understanding the properties of spaces through simpler or more manageable structures. For example, the circle S^1 can be viewed as a covering space of the line \mathbbR when considering the exponential map. In this case, the covering space wraps around the base space infinitely, illustrating how covering spaces can reveal the underlying structure of topological spaces.