A covering map is a continous onto map

p : C X

with C and X being topological spaces, which has the following property:

to every x in X there exists an open neighborhood U such that p ^-1(U) is a union of mutually disjoint open sets U^~_i (where i ranges over some index set I) such that p restricted to U^~_i yields a homeomorphism from U^~_i to U for every i in I.

We say C is a covering space of X.

Our 2005 contest logo portrays a complicated case of a covering space C. We will try to explain here a simpler example.

On the picture above C = R is potrayed as a spiral around a vertical cylinder, and X = S1 lies in a horizontal plane perpendicular to the axis of the cylinder.

Consider X being the unit circle S¹ in R².

Then the map

p : R S¹

with

p(t) = (cos(t),sin(t))

is a covering map, and C = R is a covering space of X = S¹.

In this case C = R can be drawn as a spiral:

The picture on the right explains the details.