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As always Wikepedia says it better than we can put together it a short time.
Overall, this is a very interesting area of math. This area covers the manifolds
which you asked in an earlier post on relativity. They tie in pretty well.
Topology is a branch of mathematics concerned with spatial properties preserved under bicontinuous deformation (stretching without tearing or gluing); these are the topological invariants. From around 1925 to 1975 it was the most important growth area within mathematics.
It has often been said that a topologist is a person who cannot tell a donut from a coffee cup with a handle — because both are solids with a single hole. Topology has sometimes been called rubber-sheet geometry, because it does not distinguish between a circle and a square (a circle made out of a rubber band can be stretched into a square) but does distinguish between a circle and a figure eight (you cannot stretch a figure eight into a circle without tearing). The spaces studied in topology are called topological spaces. They vary from familiar manifolds to some very exotic constructions.
After you jumped into Calculus, you first saw a real axis (a set) and the functions defined on it. You were instructed to do all kinds of analysis, like limits, derivatives, integrations, etc. Have you thought about why it has to be a "real axis" and the "functions" defined on it? If you have ever had a chance to learn complex analysis, you have seen that it really doesn't have to be the "real axis", it could be a complex plane. The functions don't have to be those defined along the real axis, they could be something with complex variables.
Actually, you have met a lot of special sets (compared to real axis) and functions, which you may not have been aware how special they are. How about the determinant of square matrices:
f(A)=det(A)?
It is so natually defined, and the "set" consists of all square matrices, instead of the real numbers. Have you thought about the integration:
F(f)=Integrate[f(x),(a,b)]? -- integration of f(x) over (a,b)
Then the "set" consists of all integrable functions.
You can find all kind of examples.
Now, comes the real question: what are the essential part after peeling off those distractive nusances?
A set and a bunch of functions? Yes, they are the important parts, but you need one more thing so that you can define the basic property of a function --- continuity. That is "topology".
Here is one simple example. Assume that you have a set A={a,b}, and a function, f:
f(a)=0,
f(b)=1.
There is no way to tell if the function is continuous or not, because we don't have enough infomation. That's where the "topology" fits in. Actually, "topology" boils down to the defintion of "open sets".
topology 1: open set {a}, {b}, {a,b}, {empty}, then the function f is continuous;
topology 2: open set {a,b}, {empty}, then the function f is discontinuous.
The set A plus the "topology" produces a "topological space". Yes, we have just constructed two "topological spaces" based on the same set and two different topology.
After you construct the topological space, you may not be satisfied with only limit operation, you may want to do some operation like differetiaion and integration. Usually, you cannot do it globally, For example, if you have a function f(x) defined only on a sphere, then usually you cannot do f(x+Δx), because x+Δx usually falls off the sphere. There is one solution to this. You take off a patch from the sphere and flatten it in the plane. Then all those operation can be carried out at least locally. This kind of topological spaces are called "manifolds".
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