The definition of topological spaces is very general and allows for some rather unusual spaces that have properties quite different from R^n. Put another way, there are some intuitive properties one might expect of a topological space which turn out to not necessarily be true from definition.
For example, there is nothing in the definition which says that two points can't be in exactly all the same open sets. However, two points in exactly all the same open sets are topologically indistinguishable from one another. Topologically, they are the same point.
An example is the space {a,b,c} with the topology {{}, {a,b}, {a,b,c}} where there is no topological distinction between a and b.
Even though the definition allows it, we will be restricting ourselves to topological spaces where every distinct pair of points is topological distinct. In other words, for any two points, there is an open set containing one but not the other. Such spaces are called T_0 spaces.
Furthermore, we will be dealing with spaces such that, for any two points, each is in an open set that the other one isn't. Such spaces are called T_1 spaces. The definition sounds very similar to T_0 but it is slightly more restrictive. T_0 only requires that one of the points is in an open set the other one isn't. T_1 requires this to be true of both points in the pair. Clearly all T_1 spaces are T_0 spaces.
A space that is T_0 but not T_1: {a,b} with the topology {{}, {a}, {a,b}}.
A further restriction turns out to be necessary in order to guarantee some of the intuitive characteristics of things like the real numbers.
In a T_1 space, we require that for any two points x and y, x is in an open set that y isn't (let's call it U) and y is in an open set that x isn't (let's call it V). There is nothing that says U and V are disjoint. They can intersect (as long as neither x nor y are in that intersection).
If a disjoint U and V exist for each pair of points, then we have what is called a T_2 space.
It may seem an arbitrary restriction to go from a T_1 to a T_2 but it turns out that this additional requirement is what allows us to define a metric on a space or take unique limits of sequences.
The additional axioms defined for T_0, T_1 and T_2 spaces say something about how separated the points have to be. For this reason they are referred to as separation axioms. There are more (T_3, T_4, etc) but, for our purposes, it is the T_2 axiom (and the T_1 and T_0 axioms that it implies) that are important to us.
T_2 spaces are also called Hausdorff spaces. From this point, pretty much all the topological spaces we deal with will be T_2/Hausdorff spaces. One cute way to remember the meaning of Hausdorff spaces is to think that for any pair, each point as "housed-off" from the other (i.e. is in an open set disjoint from at least one of the open sets the other point is in).
UPDATE: next post
The original post was in the category: poincare_project but I'm still in the process of migrating categories over.