We owe the drawing of the Tshirt design and our 2005 contest
logo to professor Larry Smolinsky from LSU Math Department.
It portrays a covering space.
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 = S^{1} lies in
a horizontal plane perpendicular to the axis of the
cylinder. 
Consider X being the unit circle S^{1}
in R^{2}.
Then the map
p : R
S^{1}
with
 p(t) = (cos(t),sin(t))
is a covering map, and C = R
is a covering space of X = S^{1}.
In this case C = R
can be drawn as a spiral:
The picture on the right explains the details.
