Topology is the study of the properties of spaces that are insensitive to continuous deformations. Physicists recognize several different subdivisions of topology. 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 continuous 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 continuous functions is 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 continuous 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 not equivalent. 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 that the corresponding spaces are not equivalent.
As an example consider again the sphere and the donut. The critical feature characterizing their topological inequivalence is the existence of non-contractible 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 non-contractible circles, mathematicians say the first homology group of the donut is two-dimensional, whereas the first homology group of the sphere is zero-dimensional. These homology groups are among the simplest algebraic objects 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 essentially count the number of inequivalent n-dimensional non-contractible cycles that exist in the topological space. The dimensions of all the homology groups are called the betti numbers. 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 ten-dimensional space consisting of our usual four dimensions of space and time and a tiny six-dimensional space attached to each point. The topology of this six-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 indispensable in understanding such notions as defects, vortices, and solitons.
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