What are the main differences in approach between standard? topology and algebraic topology?
Lonewolf here is something amazing:
It is a map of maths.
It is multilevel clickable. You click on topology and it gives you
a list of different branches of topology, including algebraic, then you click on algebraic and it tells you what it is
it also tells some history in some cases, or what it might be good for (always a problem)
Great link, Marcus. Thanks. It's much better than mathworld's superficial explanations.
That site is called "The Mathematical Atlas----a clickable index map of mathematics"
the actual answer to your question might be less fun than using this map to find it
I will try to answer for my own moral and spiritual improvement
(since you can get the answer by yourself without my help)
what you call "standard? topology" is ordinarily called
"general topology"----sometimes "point-set toplogy"
The structure you study is very simple----a set X and a collection
T (called the "topology" of X) of subsets, which gives a vague notion of proximity or neighborhood without having to use a yardstick or tapemeasure
So (X,T) is a topological space, the basic space X together with this collection of subsets----which satisfy some very simple-sounding axioms and embody a notion of neighborhood.
For no particularly good reason the members of T are called the
"open" sets. Well if you get into it I guess there is a good reason to call them that.
Also an idea of limit---the limit of a sequence of points in X---can be defined using the open sets----i.e. the sets of the topology of X.
There are no gizmos or algebraic machinery or Lie groups. The basic axioms are beautifully simple and they start you off on a fun trip that can last a semester or a couple of hundred pages.
Algebraic topology is much fancier than plain old general topology. You learn methods of constructing algebraic objects on your basic topological space (X,T), and these objects extract or contain information about the underlying space.
Also the space may be given additional structure like a metric or a differential structure in order to facilitate defining algebraic objects---oh yes, and "fiber bundles" as if that were not already enough. So then algebraic objects arise upon the fair face of the space X and these gizmos enable you to discover stuff about the underlying space. Sometime the algebraic doodad is called an "index" which means that it indicates something about X. Or it may be called a "characteristic" (as in Euler characteristic) which means that it characterizes something about X. If the original index or characteristic is a mere NUMBER then the algebraic topologist is unhappy unless he can generalize the number into a group or something even better. And so one gets things like "homotopy group", and many things named after Foreigners. Like the "So-and-so Index". I always liked the sound of Atiyah-Singer and Grothendieck (a great name)
Grothendieck was one of the most creative mathematicians of the past century and he also was bald and played a gangster in a wonderful 1961 French movie called "Zazie dans le Metro". Or else he was the tough-guy bartender. Algebraic topologists will go to see Zazie only for a chance to see Grothendieck, which is why the film is still shown periodically in Art movie theaters.
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