Manifold #1. Point Set Topology and Topological Spaces매니폴드 #1. 포인트 셋 토폴로지와 토폴로지컬 스페이스

This is the first post of the Manifold series, which continues my notes on XylyXylyX’s lectures after the Tensor series. The Tensor series dealt with coordinates and vector spaces. Here we build up the manifold, the mathematical basis of coordinate systems, starting from sets and topology.

These are my notes on XylyXylyX’s What is a Manifold? lectures. This post covers lesson 1 of the playlist, Point Set Topology and Topological Spaces.

Why start with topology?

The final goal of this series is the manifold. A manifold is the mathematical foundation of the idea of a coordinate system, and coordinate systems loom large in general relativity. We keep the rigor at an intermediate level, and before reaching the manifold we first survey point set topology. Physics students often learn differential equations and linear algebra but never study topology separately. It still helps for analysis, so we will pick out only the key points.

A set of points with no structure

We use two examples throughout the series. One is the two-dimensional plane $\R^2$ we all know, and the other is the open interval $(0, 1)$ on the real line. The open parentheses mean the endpoints $0$ and $1$ are not included.

For now these are just sets with a huge number of points. They are not vector spaces, so points cannot be added and there is no zero point. Each point may have a name (a coordinate), but the name is only a name. We will give them structure, and first we say there is a function $d$ that measures the distance between two points. This distance function is a different thing from the metric we met in vector spaces. It only tells us how far apart two points are. Let us name these sets: the plane is $X$ and the interval is $Y$.

What is a topology?

On a set $X$ we put the following structure. Make a set whose members are subsets of $X$, and call it the topology $\mathcal{T}_X$ of $X$. Each subset in this collection is called an open set. That is, an open set is an element of the topology.

Not every collection is a topology. There are rules.

1. The empty set $\emptyset$ is an element of $\mathcal{T}_X$.
2. The set $X$ itself is an element of $\mathcal{T}_X$.
3. The union of open sets is an open set, whether two of them or infinitely many.
4. The intersection of open sets is an open set, but only for finitely many.

If we find a collection that satisfies these four conditions, we have given $X$ a topology. Which open sets we choose is up to us, and as long as the conditions hold it is a legitimate topology.

Let us check by hand. In a set with three elements, pick the subsets you want to call open. The four conditions are tested one by one. When a union or an intersection is missing from the collection, the toy tells you which set is missing.

Q. The set is finite, so why do we not check infinite unions separately?
A. In a finite set, even if we combine infinitely many subsets there are only finitely many distinct subsets in the end, so repeating the check two at a time is enough. That is why the toy above checks only two at a time.

A topology on the plane: open balls

For open sets on the plane we use open balls. The open ball with center $p$ and radius $r$ is this set.

The inequality is $<$, so the boundary is not included. On the plane it is a disc without its boundary, in three dimensions a ball, and the same idea generalizes to higher dimensions. If we declare every open ball, with any center and any radius, to be an open set, we already have infinitely many open sets.

But open balls alone will not do. By rule 3, the union of two open balls must also be an open set. Taking unions of more open balls gives almost any shape we can make in two dimensions. Even a square without its boundary can be made by filling it with ever more balls. Intersections work the same way: the overlap of two balls is an open set.

Then is there anything that is not an open set? A single point is one. An open ball cannot have radius 0, and by rule 4 intersections of open balls are allowed only finitely many times, so we can never shrink down to a single point. An infinite intersection could narrow down to a point, but we did not allow it.

Since unions of open balls can build any open set, the open balls are called a base of this topology. A base is a subset of the topology whose members can be united to make every element of the topology. If we take all rectangles without boundary as the base, we get the same topology. This topology of the plane is called the usual topology. The usual topology of the line has all open intervals as its base.

Other topologies

The same set can be given other topologies.

• Discrete topology: every single point is an open set. By rule 3 every subset becomes an open set, so the topology is the power set, the collection of all subsets, and it is the largest topology possible.
• Trivial topology: the only open sets are $\emptyset$ and $X$. It satisfies all the rules, but it is too poor to be of much use.

Both can be given to any set. Try them with the “Discrete topology” and “Trivial topology” buttons in the toy above. The useful topologies lie in between.

The interval $(0, 1)$ has one more topology worth noticing. For the natural numbers $n = 2, 3, 4, \dots$, declare all the open intervals $\left(0,\ 1 - \tfrac{1}{n}\right)$ to be open sets, and also include $\emptyset$ and the whole interval $(0, 1)$. The right ends of these intervals are $\tfrac12, \tfrac23, \tfrac34, \dots$, approaching $1$ as closely as we like. The union of two intervals is the larger one and the intersection is the smaller one, so all rules hold. Even though the topology has infinitely many elements, they can be counted by $n$, and in this case the base and the topology are almost the same. It is a different topology from the usual one. The plane has topologies other than the open ball one too, but they are very subtle, so I will deal with them when I use one as a counterexample later.

Topological spaces

Finally, one term. A set $X$ together with its topology $\mathcal{T}_X$, bundled as a pair, is called a topological space. Do not understand the word topological space as a shape such as a donut or a sphere. Those belong to geometric topology, differential topology, and algebraic topology. In the point set topology we are doing now, a set and a topology on it are everything. The same set has several topologies, and a different topology gives a different topological space.

In the next lecture we look at properties that tell topologies apart, namely connectedness, countability, and separability, using the interval as an example, and then check which properties the plane with its usual topology has.