Topology is the study of those properties of geometric
configurations which remain invariant when these configurations
are subjected to one-to-one bicontinuous transformations, or
homeomorphisms (see Chapter XII, § 3). We call such properties
topological invariants. For example, the property of a circle to
separate the plane into two regions is a topological invariant;
if we transform the circle into an ellipse or into the perimeter
of a triangle, this property is retained. On the other hand, the
property of a curve to have a tangent line at every point is not
a topological property; the circle has this property but the
perimeter of a triangle does not, although it may be obtained
from the circle by means of a homeomorphism. As can already be
seen from the above example topology operates with more general
concepts than analysis; differential properties of a given
transformation are nonessential for topology, but bicontinuity
is essential. As a consequence, topology is often suitable for
the solution of proble