Why topology on a set is defined the way it is?

[Ahmed Abdullah]

Following is from Wolfram Mathworld

"A topological space, also called an abstract topological space, is a set X together with a collection of open subsets T that satisfies the four conditions:

1. The empty set is in T.
2. X is in T.
3. The intersection of a finite number of sets in T is also in T.
4. The union of an arbitrary number of sets in T is also in T. " http://mathworld.wolfram.com/TopologicalSpace.html
   

My question is why topology on a set is defined in this way? How these definition connect with our intuitive idea of different kind of topological space (i.e line, graphs, 3-sphere, torus etc).

I am obviously new in topology and will be glad if you explain in basic term. I'd be specially interested to know how one differentiate between a straight line segment and a "Y" shaped graph using these definitions.

I have convinced myself of one way, please let me know if it is correct. I can separate Y naturally in three segment let's name them a,b and c.

Let X={a,b,c}.

So the topology Y on X will be { {},{a,b,c},{a,b},{a,c},{b,c},{a},{b},{c}}.

We can break a line segment on three part. Let's do likewise for line segment l.

So the topology l on X will be { {},{a,b,c},{a,b},{b,c},{b}}. Topology Y and l are on X and obviously different. :)

 

[Svein]

My question is why topology on a set is defined in this way?

In mathematics things are defined the way they are because somebody found it a useful way of handling several (often unrelated) problem sets.

 

[Stephen Tashi]

How these definition connect with our intuitive idea of different kind of topological space (i.e line, graphs, 3-sphere, torus etc).

The definition you gave pertains to the subject called "General Topology". General Topology does not provide enough definitions and assumptions to deal with questions about the differences between a 3-sphere and a torus. To deal with differences like that you need to study "Algebraic Topology".

General topology provides a way to treat mathematical questions involving the (vague) concept of "closeness" in a unified fashion. For example, in an elementary approach, $lim_{x \rightarrow a} f(x) = L$ requires one definition for real valued functions of a real variable and a different definition when x and L are two dimensional vectors. Taking the definition of "limit of a function" given by General Topology, the same definition of limit of a function applies to both cases. The only change that is made is that "open set" means one thing on the real line and a different thing in 2-dimensions.

 

[HallsofIvy]

The whole point of topology is to generalize our concepts of "limits" and "continuity". A "topology" on a set is the collection of all open subsets of that set. And those properties are the properties of open intervals on the real line that are used in defining limits.