Topology is the study of properties of topological spaces invariant
under homeomorphisms. See Section 2 for a precise denition of topo-
logical space.
In algebraic topology, one tries to attach algebraic invariants to
spaces and to maps of spaces which allow us to use algebra, which is
usually simpler, rather than geometry. (But, the underlying motivation
is to solve geometric problems.)
A simple example is the use of the Euler characteristic to distinguish
closed surfaces. The Euler characteristic is dened as follows. Imagine
the surface (say a sphere in R3
) triangulated and let n0 be the number