I'm beginning to wonder if I was wise to relay so heavily on point set topology to explain some things about sets. So let's do some topology here. By the way, I'm working (virtually) without sources, so if there are errors, someone shout

Consider a point set S.

Definition

A collection T of subsets of S is said to a be topology on S iff the following are true:

finite intersection of elements of T (aka subsets of S) are in T;

arbitrary union of elements of T (aka subsets of S) are in T;

S is in T;

Ø is in T.

The pair (S,T) is referred to as atopological space. (Note that it is usual to abuse notation and assert, for example, that "S is a topological space" (when it is, of course), so take note: it means S = (S,T) in context)

I'll give some examples in a minute, but let's press on a bit. The elements of T are said to be theopen setsin (S,T). Subsets of S which are in the complement of T are theclosedsets in (S,T). (the complement of any set A in T is S - A)

So let's have a couple of examples. Consider the subsets S and Ø of S. The topology T = {Ø, S} is referred to as the indiscrete (or trivial, or concrete) topology on S. Note that the complement of S is Ø, the complement of Ø is S, so these guys are both open and closed in this topology (well all topologies actually).

Now consider the power set on S, that is all elements and their combinations. (The power set P(S) for any set with n elements, has cardinality 2ⁿ e.g. for S = {a, b, c}, P(S) = {{a}, {b}, {c}, {a,b}, {a,c}, {b,c}, {a,b,c}, Ø}). This is in fact a topology on S and is referred to as the discrete topology on S: it is the finest possile topology on S. Note that for each set in T = P(S), it's complement also appears in T, so all sets in this topology are both open and closed.

In fact it is the case that any topology T on S is a subset of P(S).

Let R be the set of real numbers. In thestandardtopology on R, elements of T (aka open sets in (R,T)) are open intervals of the form (a, b). The union of all such intervals is the real line.

Let (S,T) be a topological space. For some point s in S, an open set in (S,T) containing s is referred to as aneighbourhoodof s.

Let X and Y be topological spaces (note the abuse of notation I referred to earlier). A map f: X → Y is said to becontinuousat x if, for each x in the domain of f there is a neighbourhood U of f(x) in Y whose pre-image f<sup>-1</sup>(U) is open in X. It is (relatively) easy to show that this corresponds to the usual ε - δ definition of continuity we learn in school.

Obviously, if f is a bijection, we can think of f<sup>-1</sup> as a continuous inverse. Under these circumstances, f:X → Y is referred to as ahom. Note that the above places neighbourhoods of X and Y in one-to-one correspendence. But neighbourhoods are open sets i.e. elements in the topology, so X and Y are topologically equivalent, which is the meaning of homeomorphic. It is the topological equivalent of an isomorphism (and gives rise to the tired old joke about topologists not knowing the difference between a donut and a coffee cup)eomorphism

If homeomorphic spaces are topologically equivalent this means they must share what are known as topological properties. I'l briefly mention the three that seem, to me, to be the most important.

The Hausdorff property.

Let X be a topological space. If, for anypairof points x notequal y in X there exist neighborhoods U and V of x and y respectively such that U ∩ V = Ø, X has the Hausdorff property ("is Hausdorff")

The Connected property.

If, in a topological space X the only sets both open and closed are X and Ø, X is connected. Equivalently, X is connected if it isnotthe union of disjoint non-empty open sets.

The Compact property.

A collection of subsets of a topological space X whose union is X is called a cover of X. If these subsets are all open it is an open cover. And if there is a subclass of this collection similarly covering X, it is a subcover. X is compact if every open cover has a finite subcover.

Finally, although for any set there are a number of topologies that can be placed on it, it is often of no real interest what the actual topology is. But it isalwaysnecessary to list the topological properties of the space.