I was reading about A Beautiful Mind and John Nash's contributions to mathematics and it said that he did some research on topology. Can someone explain to me basically what the mathematical study of topology is? What are its goals? I know it involves set theory somehow and that's about it. Thanks.

Hi LionHearted,

Topography is the art of making detailed maps.

A topology is a collection of all open sets of a metric space.

addition
You don't even necessarily have to deal with a metric space, it is sufficient to deal with what is called a "topological space", defined as follows (webster.com)

"Main Entry: topological space
Function: noun
Date: 1926
: a set with a collection of subsets satisfying the conditions that both the empty set and the set itself belong to the collection, the union of any number of the subsets is also an element of the collection, and the intersection of any finite number of the subsets is an element of the collection"

Bye!

Crisp

There is a short thread about Topology here (http://www.sciforums.com/showthread.php?s=&threadid=13975)

Basically, topology is the study of space but is not concerned with distances.

Let's look at 2 dimensions. Take a flat sheet of paper. It has a particular topology. Bend the sheet any way you like and you won't change the topology. Stretch the sheet and the topology also stays the same. So, for example, a bowl-shaped sheet has the same topology as a flat sheet.

Similarly, a coffee cup with a handle has the same topology as a donut, because in principle one shape can be turned into another by stretching. However, a donut has a different topology from a flat sheet because no amount of stetching of a sheet will turn it into a donut shape; you'd need to put a hole in the sheet to do that.

Originally posted by Crisp
Hi LionHearted,

Topography is the art of making detailed maps.

A topology is a collection of all open sets of a metric space.

addition
You don't even necessarily have to deal with a metric space, it is sufficient to deal with what is called a "topological space", defined as follows (webster.com)



i would like to stress that topology concerns itself with those properties of a space that do not depend on the metric. or with spaces which do not allow metrics. as soon as you start talking about properties that depend on the metric, then that discussion belongs properly in geometry or analysis.

it is true that a topology can be defined in terms of a metric, as you mentioned, Crisp. the set of open epsilon neighborhoods in the metric do form a topology. but this is not the most useful definition, because it obscures the fact that different metrics sometimes generate the same topology. and some pathological spaces do not admit a metric at all.

intuitively, what it means to say topological properties depend only on the topology, and not the metric, is like what james described. a topology on a space is basically a rule that tells you which points are near which other points. it does not concern itself with how near they are, that is what a metric is for. so if you stretch or bend the space (thinking of it as a rubber sheet), you change the metric, but not the topology. you can stretch a disk into a hemisphere, but not into a sphere, not without altering the topology. so the disk and the hemisphere are homeomorphic (that s lingo for topologically equivalent), but the disk and the sphere are not.

this is my paraphrasal of the second, more correct, more abstract definition that crisp gave.

What are the applications of topology?

algebraic topology is used quite a bit in string theory, although its not clear that string theory itself has any applications.

but you know, the question "what are the applications?" has always been a turn off for me. i think mathematics is worth doing for its own sake.

One application is knot-tying. Topology describes all kinds of knots. Just recently, a book was published on the "85 ways to tie a tie" (85 may be the wrong number). Interesting stuff.

Originally posted by lethe
but you know, the question "what are the applications?" has always been a turn off for me. i think mathematics is worth doing for its own sake.

I agree. I learn mathematics just for the joy of it. I just wanted to know the applications to get a better idea of what the theory is.