You might have heard the expression that to a topologist, a donut and
a coffee cup appear the same. In many branches of mathematics, it is
important to define when two basic objects are equivalent. In graph
theory (and group theory), this equivalence relation is called an
*isomorphism*. In topology, the most basic equivalence is a
homeomorphism, which allows spaces that appear quite different in most
other subjects to be declared equivalent in topology. The surfaces of
a donut and a coffee cup (with one handle) are considered equivalent
because both have a single hole. This notion needs to be made more
precise!

Suppose
is a bijective (one-to-one and onto)
function between topological spaces and . Since is
bijective, the inverse exists. If both and are
continuous, then is called a *homeomorphism*. Two topological
spaces and are said to be *homeomorphic*, denoted by
, if
there exists a homeomorphism between them. This implies an
equivalence relation on the set of topological spaces (verify that the
reflexive, symmetric, and transitive properties are implied by the
homeomorphism).

Be careful when mixing closed and open sets. The space is not
homeomorphic to , and neither is homeomorphic to . The
endpoints cause trouble when trying to make a bijective, continuous
function. Surprisingly, a bounded and unbounded set may be
homeomorphic. A subset of
is called *bounded* if there exists a ball
such that
. The mapping
establishes that
and
are homeomorphic. The mapping
establishes that and all of
are
homeomorphic!

The bijective mapping used in the graph isomorphism can be extended to
produce a homeomorphism. Each edge in is mapped continuously to
its corresponding edge in . The mappings nicely coincide at the
vertices. Now you should see that two topological graphs are
homeomorphic if they are isomorphic under the standard definition from
graph theory.^{4.3} What if the graphs are not isomorphic? There is still a
chance that the topological graphs may be homeomorphic, as shown in
Figure 4.2. The problem is that there appear to be
``useless'' vertices in the graph. By removing vertices of degree two
that can be deleted without affecting the connectivity of the graph,
the problem is fixed. In this case, graphs that are not isomorphic
produce topological graphs that are not homeomorphic. This allows
many distinct, interesting topological spaces to be constructed. A
few are shown in Figure 4.3.

Steven M LaValle 2012-04-20