Good Math, Bad Math

Topology usually starts with the idea of a metric space. A metric space is a set of values with some concept of distance. We need to define that first, before we can get into anything really interesting.

Metric Spaces and Distance

What does distance mean?

Let's look at a set, S, consisting of elements s₁, s₂, s₃,...,s_n. What does it mean to measure a distance from s_i to s_j?

We'll start by looking at a simple number-line, with the set of real numbers. What's the distance between two numbers x and y? It's a measure of how far over the number-line you have to go to get from x to y. But that's really circular; we've defined distance as "how far you have to go", which is defined by distance. Let's try again. Take a blank ruler, and put it next to the numberline, so that the left edge of the ruler is on X, and then draw a mark on the ruler where it touches Y. The length of the ruler up to that mark is the distance from x to y. The reason that this one isn't circular is because now, you can take that ruler, and use it to answer the question: is the number v farther from the number w than x is from y? Because the ruler gives you a metric that you can use that is separate from the number-line itself.

A metric over S is a function that takes two elements s_i and s_j, and returns a real number which measure the distance between those two elements. To be a proper metric, it needs to have a set of required properties. To be formal, a function d : S × S → ℜ is a metric function over S if/f:

1. ∀ s_i, s_j ∈ S: d(s_i,s_j) = 0 if/f i=j. (the identity property)
2. ∀ s_i, s_j ∈ S: d(s_i,s_j) = d(s_j,s_i) (the symmetry property)
3. ∀ s_i, s_j, s_k ∈ S: d(s_i,s_k) ≤ d(s_i,s_j) + d(s_j,s_j) (the triangle inequality property)

Some people also add a fourth property, called the non-negativity property; I prefer to leave it out, because it can be inferred from the others. But for completeness, here it is: ∀ s_i, s_j ∈ S: d(s_i,s_j) ≥ 0.

A metric space is just the pair (S,d) of a set S, and a metric function d over the set.

For example:

1. The real numbers are a metric space with the ruler-metric function. You can easily verify that properties of a metric function all work with the ruler-metric. In fact, they are are all things that you can easily check with a ruler and a number-line, to see that they work. The function that you're creating with the ruler is: d(x,y) = |x-y| (the absolute value of x - y). So the ruler-metric distance from 1 to 3 is 2.
2. A cartesian plane is a metric space whose distance function is the euclidean distance:
   d((a_x,a_y), (b_x,b_y)) = ((a_x-b_x)² + (a_y-b_y)² )^1/2.
3. In fact, for every n, the euclidean n-space is a metric space using the euclidean distance.
4. A checkerboard is a metric space if you use the number of kings moves as the distance function.
5. The Manhattan street grid is a metric space where the distance function between two intersections is the sum of the number of horizontal blocks and the number of vertical blocks between them.

Open and Closed Sets in Metric Spaces

You can start moving from metric spaces to topological spaces by looking at open sets. Take a metric space, (S,d), and a point p∈S. An open ball B(p,r) (a ball of radius r around point p) in S is the set of points x such that d(p,x) < r. (note: originally, this paragraph contained a major typographic error: I wrote ≤ rather than <, which causes a significant change in meaning. As usual, thanks to the commenter who caught it!)

Now, think of a subset T of S. A point p ∈ S is in the interior of T if/f there is some r where B(p,r) ∈ T. T is an open subset of S if every element of T is in its interior. A subset of space formed by an open ball is always an open subset. An open subset of a metric space S is also called an open space in S.

Here's where we can start to do some interesting things, that foreshadow what we'll do with topological spaces. If you have two open spaces T and U in a metric space S, then T ∪ U is an open space in S. So if you have open spaces, you can glue them together to form other open spaces.

In fact, in a metric space S, every open space is the union of a collection of open balls in S.

In addition to the simple gluing, we can also prove that every intersection of two open subsets is open. In fact, the intersection of any finite set of open subsets form an open subset. So we can assemble open spaces with all sorts of bizarre shapes by gluing together collections of open balls, and then taking intersections between the shapes we've built.

So now, think about a subspace T of a metric space S. We can say that a point p is adherent to T if for all r > 0, B(p,r)∩T≠∅. The closure of T (usually written as T with a horizontal line over it; sometimes written as T^* by computer scientists, because that's the closure notation in many CS subjects). is the set of all points adherent to T. (note: a typo was corrected in this paragraph. Thanks to the commenter who caught it!)

A subset T of S is called a closed subset if/f T=T^*. Intuitively, T is closed if it contains the surface that forms its boundary. So in 3-space, a solid sphere is a closed space. The contents of the sphere (think of the shape formed by the air in a spherical balloon) is not a closed space; it's bounded by a surface, but that surface is not part of the space.