What are sets made up of in topology?

Question

I am looking for a very basic answer, I am in 10th grade and studying it for a project. I don't understand what the sets are referring to. Is it nodes? Edges? Axioms? Vertices? Or does it just depend?

Answer 1

A set, in mathematics, is just a bunch of things. A set could be a bunch of numbers like $\{1, 2, 5, 1000\}$, or it could be a bunch of points on the surface of a sphere like $\{(x, y, z) : x^2 + y^2 + z^2 = 1\}$, or it could be a bunch of words like $\{\mbox{cat},\mbox{dog},\mbox{Ouagadougou}\}$.

Perhaps most confusingly, a set can be a bunch of other sets - for example, $\left\{ \left\{1, 2, 5, 1000\right\}, \left\{(x, y, z) : x^2 + y^2 + z^2 = 1\right\}, \left\{\mbox{cat},\mbox{dog},\mbox{Ouagadougou}\right\} \right\}$ is a valid, if not particularly useful, set.

A subset is a bunch of things inside a larger^[1] set. For example, some of the subsets of $\{1, 2, 5, 1000\}$ are $\{1, 5\}$, $\{2\}$ and $\{1, 2, 5, 1000\}$ itself. Additionally, the set containing nothing, $\{\}$, sometimes written as $\varnothing$, is also a subset of that set - and, indeed, of every set in existence.

A topology is a way of defining which elements of a given set are somehow "close to" or "next to" or "linked to" each other, by listing out the subsets of the set that define "neighbourhoods" of associated points. The only rules for a topology are:

1. It must contain the empty set, as well as the original set.

2. If you take any combination of subsets from the list, and find all the elements they have in common (called the intersection of the sets), then the resulting set is also in the list.

3. If you take any combination of subsets from the list, and find all the elements they cover in total (called the union of the sets), then the resulting set is also in the list.

The simplest topology, and one that exists for any set, is the trivial topology consisting of the empty set and the set itself. So $\left\{ \varnothing, \left\{ \mbox{cat},\mbox{dog},\mbox{Ouagadougou}\right\} \right\}$ is a valid topology for the set $\left\{\mbox{cat},\mbox{dog},\mbox{Ouagadougou}\right\}$.

The other simple topology that exists for any set is the discrete topology, which is one that lists all possible subsets. For example, the discrete topology on $\{A, B, C\}$ is $\left\{ \varnothing, \{1\}, \{2\}, \{3\}, \{1, 2\}, \{1, 3\}, \{2, 3\}, \{1, 2, 3\}\right\}$.

If we take a more geometric view, then the elements of a topology are called the "open sets", and if your starting set is $\mathbb{R}$, which is the set of real numbers (or essentially the number line), then you can define your open sets as being the open intervals (i.e. ranges $(x, y)$ with $x \leq y$ that don't contain their endpoints) and saying that open sets are those that can be constructed from unions of those intervals. Similarly, you can define topologies on 2D or 3D space by starting with circles and spheres and building from there.

Somewhat nicely, we can look at some topological spaces (that is, sets with topologies on them) and say that they're equivalent - if we map the elements of one set to the elements of the other, then their topologies map in the same way. This is where the whole "a donut is the same as a coffee cup" stuff comes from - by defining the right topology, we can assign characteristics to certain types of sets, especially ones that have some kind of geometry involved, along with certain "valid" ways to reshape those sets, that lets you group them together by those common characteristics. For example, you can formalise the idea of a shape with a hole in it versus one without, such that we would say that any shapes with a single hole are somehow "topologically equivalent" to each other. We can even go so far as to start talking about joining topologies together or cutting them apart or "gluing" points together and looking at how their properties behave (for example, if you take the closed interval $[0, 1]$ and just say that for your purposes the points at $0$ and $1$ are joined to each other, you have a shape that is topologically equivalent to a circle).

Which then gets to graphs - in mathematics, a graph is a set of points (the nodes or vertices) along with a set of pairs of points (the edges). It doesn't necessarily have geometric properties - a database of Facebook users (nodes) and information on which ones are friends (edges) is a graph, as is a database that links people (nodes) to websites (also nodes) via their user accounts (edges). However, we can add some geometry by treating the edges as line segments, and that lets us look at how they relate to other things. For example, you can represent polyhedra as graphs (literally just taking their vertices and edges), and hence you can define a bunch of topologically equivalent polyhedra based on them having the same graph. So when you're working in this kind of space, the set that's of interest (topologically speaking) is a combination of the vertices and their edges, not as physical points but as discrete identities and information about which ones are joined to each other (and often also some information about the order in which they're joined, because that helps define the faces of the shape and some other handy properties).

tl;dr Yes?