Topology
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   Topology is the study of the properties of spaces that are insensitive
to continuous deformations.  The study of topology comes in different
flavors.  The simplest is called point set topology which is needed to
define the notion of a topological space itself.  A topological space
is a set of points, possibly continuously infinite in number, which is
endowed with a topology, or a collection of open sets that in essence
specifies which points in the space are close to each other.  This is
the minimum mathematical structure needed to define continous
functions from one topological space to another.  Continuous functions
are simply those functions that take points that are near each other
in one space to points that are near each other on the target space.
An example of a family of continous functions are polynomials in one
variable which map the real line back to itself.

   Two topological spaces are considered equivalent if either can be
deformed into the other in a continous way without making any rips in
each space along the way, or without plugging up any holes.  For
example, a hollow sphere with a hollow handle attached to it is
equivalent to a hollow torus or donut, but the sphere without the
handle attached is not equivalent.  No matter how hard one tries one
cannot deform a sphere into a donut. One branch of topology called
algebraic topology is especially useful in physics and enables one to
characterize when two topological spaces are inequivalent.  The basic
idea is to attach to each topological space an algebraic object.  Two
topological spaces will be equivalent only if the corresponding algebraic
objects are the same.  Thus if one computes the algebraic objects corresponding
to two different spaces and finds that they are different, one has essentially
proved the inequivalence of the corresponding spaces.

   As an example consider again the sphere and the donut.  The critical
feature characterizing their topological inequivalence is the existence
of noncontractible circles on the donut.  Consider for example any
circle on the sphere.  It can always be contracted to a single point, just
like a lasso can slip off any round object.  However the hollow donut has
two inequivalent circles that cannot be contracted to a point.  One circle
travels around the equator of the donut.  The second circles around 
one end of the donut, in much the same way one's thumb and index finger
would curl around the donut when holding it at one end. 

   Since the donut has two noncontractible circles, mathematicians say
the first homology group of the donut is 2 dimensional, whereas the
first homology group of the sphere is 0 dimensional.  These homology
groups are among the simplest algebraic obejcts that can be associated
to topological spaces.  This picture can be generalized to higher
dimensional objects, and n-dimensional homology groups can be defined,
which essentally count the number of inequivalent n-dimensional
noncontractible cycles that exist in the topological space.  The
dimensions of all the homology groups are called the betti numbers and
a pivotal result in the field of algebraic topology is that if the
betti numbers of two different spaces differ then the two spaces cannot
be deformed into each other.

   Topology shows up in a crucial way in the modern theories of physics.
For example string theory presents a picture of the world as a 10 
dimensional space consisting of our usual 4 dimensions of space and time
and a tiny 6 dimensional space attached to each point.  The topology
of this 6 dimensional space is highly constrained by consistency 
requirements in physics, and the constraints are naturally expressed
in terms of something similar to the betti numbers described earlier.
In condensed matter theory as well topology has proved a useful tool in
characterizing classical field configurations and is indispensible 
in understanding such notions as defects, vortices and solitons.