The Motivation

Topology is an incredibly useful and important branch of mathematics. It is, in essence, the study of spaces, in the most general sense of the term. We start by looking at some arbitrary collection of points. This immediately puts us into the mathematical realm of set theory. It is assumed here that the reader knows the meanings of intersections, unions, and complements of sets, among other basic set theoretical concepts.

So we have this arbitrary collection of "points". It could be finite or infinite; countable or uncountable. There need be no notion of "distance" between points; one point might mean "apples" and another might mean "oranges". The point is, this structure is meant to work on an arbitrary set. The question we ask ourselves is: On a set of points, X, what is the minimal structure that we need to introduce to talk about various familiar ideas like "continuity", "limits", or "nearness" of a point to a collection of points?

Open sets in the Real Numbers

Let's start with a situation we understand well, and try to rephrase our understanding more generally. Specifically, we know the meaning of "continuity", "limits" and "nearness" on the real numbers. These concepts are all defined rigorously in the realm of calculus on the real numbers.

So let's start with the x-y plane, X = R2. On the x-y plane there exists the useful idea of "open sets". This can be defined in terms of a distance function, d, where

d(x,y) = √((x1-y1)2 + (x2-y2)2)

Then we can introduce the concept of open balls of radius r centered on p

Br(p) = {q:d(p-q) < r}

In other words, an open ball of radius r at p consists of all points q such that the distance from p to q is less than r. Basically, all points within a distance r from p. Note that this definition does not include points at a distance of exactly r from p. If we wanted to also include points at exactly a distance of r from p, we would call such a set the closed ball of radius r at p.

Now we are ready to define the concept of an open set:

An open set, U, is any subset of X such that, for any point p in U, there exists an open ball at p which is entirely conained in U.

Thus, if you have a set which contains a point on its border, like the closed ball of radius r, this cannot be an open set, for if we take p to be a point on its border, there is no open ball around p which is entirely contained in the set; there will always be some part of the open ball outside the border.

Now we notice some basic properties of open sets in R2:

This is true trivially, since our definition begins with for any point p in U, and since there are no points in the empty set, it satisfies the conditions for an open set vacuously.

• The entire set R2 is an open set.

Look back at our definition. For any point p in R2, you could choose an open ball of any radius, and it will be entirely contained in R2.

• If A and B are two open sets, their union is an open set.

This is true, since if p is in the union of A and B, then it must be in at least one of these sets. For the sake of argument, let's say p is inside A. Then, since A is an open set, we can use our definition, which says there exists some open ball centered at p which is entirely contained in A. Logically, if the ball is entirely contained in A, then it must also be entirely contained in the union of A and B. Thus, this satisfies our definition for any point p (If p is inside of B instead of A, we can easily repeat the argument with a change of labels).

• If we have an infinite collection of open sets, their union is an open set.

This is true by basically the same reasoning as above.

• If A and B are open sets, their intersection is an open set.

Looking back at our definition, if p is contained in the intersection between A and B, it must be contained in both A and B. Therefore, there exists a ball of radius rA centered at p which is entirely contained in A, and there exists a ball of radius rB which is entirely contained in B. We know that one of these radii is greater than or equal to the other, so without loss of generality, we'll say rA < rB. Then the ball of radius rA is entirely contained in the ball of radius rB, which is entirely contained in B. Thus, the ball of radius rA is entirely contained in both A and B, and is therefore contained in their intersection.

• For any finite collection of open sets, their intersection is open.

This can be shown by iterating the last property over and over; if A, B, and C, are open, then (A B) and C are each open, then A B C is open.

Note that infinite intersections are not necessarily open. For example, if we took an infinite number of open balls (centered at the origin) of radius equal to (1 + 1/n), for n = 1, 2, 3... then their intersection would be the closed ball of radius 1 (i.e. the points which are contained in all open balls of radius (1+ 1/n) are the points greater than or equal to a distance r from the origin). The closed ball of radius 1 is not an open set. Thus, only finite intersections are necessarily open.

So, we have learned that the collection of open sets in R2 satisfies these three properties:

1. The empty set and the whole set are open.
2. Arbitrary (finite or infinite) unions of open sets are open.
3. Finite intersections of open sets are open.
We note that this discussion relied on having a distance function which gave us the definition of an open ball. The distance function defined our open sets. There is a natural and useful generalization of this structure of open sets that applies to any set of points, even one without a distance function. This is where we finally come to the concept of a topology.

Topology

Let X be any set of points. In place of the concept of "open balls" which rely on a distance function, just declare certain subsets of X to be open.

First demand, however, that this collection of open sets obey properties 1, 2, and 3 above.

Now, we define a topology, T, on X to be the collection of subsets of X which are declared to be open sets, and which satisfy 1, 2, and 3.

Intuition

How does this notion of "open sets" help us to characterize the concept of "nearness" on a space? It helps to think of open sets containing a given point, p, as "neighborhoods" of that point. If we have some idea of "nearness" of points to each other, then every neighborhood of p contains all points within a certain degree of "nearness" to p. This is why we used open balls in R2. Open balls give a definitive description of nearness to p, and thus we can say an open set containing p is a "neighborhood" of p, since it always contains an open ball about p; it always contains all points sufficiently near p. It might contain points "far" from p as well, but it definitely contains all points that are close to p, given a sufficiently strict definition of "closeness".

Sets which are not open don't work for this purpose, as we can see from our example on R2. If a set contains a point on its border, then the set is not a neighborhood of that point. There are always points very near to p that are outside the set, on the other side of the border. So, a good way of thinking of open sets (other than that they satisfy the three axioms) is that they are the sets which are neighborhoods of all their points. You can check that axioms 1-3 follow quite naturally from this point of view (in the same way that they showed up in the discussion on R2), even though it's more of an intuitive picture, and not a mathematically rigorous construction.

Examples of Topologies

Let X = R2, and let the topology simply consist of the sets we are used to calling "open". This is known as the standard topology or metric topology on R2.

Let X = the set of all different types of fruit. One point might be "pears", and another might be "bananas". We can define a topology by simply declaring every set of points to be an open set. You can check that this is a topology, in that it satisfies the axioms 1 - 3 above. It is clear that this topology can be declared on any set; it is known as the discrete topology on X. If a topology is meant to give a notion of "nearness", one could say that in the discrete topology, every point is "far" from every other point.

Again, let X = the set of types of fruit. Another topology we can simply declare is that our list of open sets consist of two sets: The empty set and X itself. Again, this topology can be defined on any set, and it is known as the trivial topology or the indiscrete topology on X. In contrast to the discrete topology, one could say in the indiscrete topology that every point is "near" every other point.

Now we apply the notion of a topology to define the concept of continuity on an arbitrary collection of points.

Continuity on the Real Line

We start in a familiar manner, again, looking at a function which takes points in R to points in R. The mathematical definition of a continuous function on R is as follows:

f is said to be continuous at a point x0 in R if for every ε > 0 there exists a δ > 0 such that |f(x) - f(x0)| < ε if |x - x0| < δ

This definition is often cryptic for anyone who hasn't seen it before, but the point of the definition is that for a sufficiently small deviation from x0 we get a small deviation from f(x0). These "small" deviations are characterized by the variables δ and ε, respectively. It's always possible to choose a δ such that the deviation in f is less than any ε.

Now we can generalize the definition in light of this new concept of a topology:

Let f be a map from X1 to X2, two topological spaces. f is said to be continuous at p if for every open set V in X2 such that f(p) is contained in V (every neighborhood of f(p)) there exists an open set U in X1, such that p is contained in U (a neighborhood of p), for which f(U) is completely contained in f(V).

If you think about this definition for awhile, thinking of "open balls" in R, you will see we get back our exact definition of continuity on R, but with no appeal to any notion of distance between points. Anyone who has worked with epsilons and deltas in proving continuity of functions should be quite happy about this result; it vastly simplifies discussions of continuity, basically by streamlining past the notion of distances. More importantly, though, we can now talk about the "continuity" of a map from, say, the set of fruits to the set of lines passing through the origin in three dimensions. All we need to know is the topologies of the two spaces.

Homeomorphisms

It is useful to be able to categorize distinct varieties of topological spaces. For example, if we have a set X with two possible topologies Τ1 and Τ2, characterized by the open sets {Ui} and {Vj}, and the only difference between Τ1 and Τ2 might be an ordering of the open sets (meaning Ui = Vj for some j corresponding to each i and vice versa), we certainly don't want to consider Τ1 and Τ2 to be distinct topologies on X. This leads us to ask, when should we consider {X, Τx} and {Y, Τy} to be the "same" topological spaces?

The answer comes from our notion of continuity. First of all, for us to compare two spaces (and consider them to be identical in some respect), we first need a bijective map (an invertible function) between them. This gives us a sort of "dictionary" which shows how points in the different spaces correspond to each other. This shows that the spaces are in one-to-one correspondence at the level of points. What we want is correspondence at the level of open sets. This can be achieved if we simply include the requirement that f maps open sets in X to open sets in Y and f -1 maps open sets in Y to open sets in X. As it turns out, this requirement is equivalent to the condition that f and f -1 are both continuous maps.

Two topological spaces X and Y are said to be "homeomorphic" if there exists a continuous, bijective map between them, whose inverse is also continuous. Such a continuous, bijective map is called a "homeomorphism".

To get an intuitive picture of what a homeomporphism does, it helps to look at the concept of topological invariants. Topological invariants are properties of a topological space which depend only on the topology, and hence are preserved by homeomorphic maps. If X and Y are homeomorphic, then the topological invariants on X and Y must agree. If two topological spaces, X and Y, differ by one topological invariant, then they are not homeomorphic.

Examples of topological invariants

Connectedness: A topological space X is called "disconnected" if X can be written as the union of two disjoint open sets. Otherwise, X is said to be "connected". Since this definition completely depends on the open sets of X, connectedness must be a topological invariant.

Compactness: X is said to be "compact" if any open cover of X admits a finite subcover. This property helps to distinguish between sets like the real line and the circle. Both sets have an infinite number of points, but we want to think of the real line as somehow "more" infinite than the circle. Compactness is a property that distinguishes the two.

Other topological invariants include the orientability of a space (most spaces you can think of are orientable, but, for example, the mobius band is not), and the genus of a surface (which is related to the number of "holes" or "handles" in a surface. Think of the difference between a sphere and a donut).

So, a homeomorphism can distort the space you're interested in, but it preserves properties like connectedness, compactness, orientability, and genus. The intuitive picture you should be getting from this is that a homeomorphism is a continuous deformation. We can perform such a deformation on a sphere and get many different types of blobs, but, for example, we could never continuously deform it into a donut, or into a mobius band, or into two spheres, or into a flat plane. Such transformations would be discontinuous, and would furthermore change certain topological invariants of the sphere.

In summary

When we define the topology of a space, we are implicitly giving the space a notion of continuity and limits. When we deal with a familiar set like the real numbers, this notion of continuity and limits can come directly from a notion of "distance" between two points. However, the notion of a topology is far more general and can be used to characterize much more abstract spaces.