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Topological Concepts

In this section, we will review some basic topological concepts that will be useful in understanding more advanced topics in mathematical physics.

This section is not based on Hassani, but rather on my own understanding of topology. This means that it won't be numbered like the other sections.

Table of Contents​

Open and Closed Sets​

Before even defining a topological space, we need some intuition about spaces in general. If we just skip to defining a topology over a set, it might be hard to grasp the concept of what topology even is. Often, in mathematics and physics, we can think of different things geometrically. For example, when we studied quantum mechanics, we say that , the group of rotations in 3D space, can be thought of as a sphere with antipodal points identified. In general relativity, we classified spacetimes based on how they are curved (e.g. positively curved, negatively curved, flat). To formalize this idea of "space", we need to define some concepts in topology. What makes equivalent to a sphere? Can we prove it?

Anyways, we will start by analyzing metric spaces, which are sets with a notion of distance. These are more intuitive to think about, since we can visualize distances in our head. Later, we will generalize this to spaces without a metric, but this is a good starting point.

Definition (open ball) In a metric space , an open ball of radius centered at a point is the set

In other words, an open ball is the set of all points that are within a distance from the point . In with the usual Euclidean metric, an open ball is just a sphere without its surface. If we change the metric, the shape of the open ball will change as well.

Definition (closed ball) In a metric space , a closed ball of radius centered at a point is the set

This is just like an open ball, but it includes the surface as well.

The next important idea is to distinguish points that are "inside" a set from those that are not. Intuitively, a point is "inside" a set if we can draw an open ball around it that is completely contained within the set.

Definition (interior) A point is an interior point of a set if there exists an open ball such that . More concretely, there exists some radius such that for all points with , we have .

The interior of a set , denoted by , is the set of all interior points of .

Interior sets follow some nice properties:

  • for any set .
  • for any set .
  • for any sets .
  • for any sets .
  • implies for any sets .

We will not prove these properties here, but they are not too hard to show.

For an example, consider the set . The interior of this set is the open interval , since any point in has an open ball around it that is completely contained within . However, the points and do not have this property, since any open ball around them will include points that are not in .

Notice that with the definition of an interior set, we were able to isolate out and as special points. Specifically, they are on the "edge" of the set. It seems as though the set is different from the set because of these two points. This leads us to the next important concept: open and closed sets.

Definition (open set) A set is open if . In other words, a set is open if all of its points are interior points.

For example, the set is open, since all of its points are interior points. On the other hand, the set is not open, since the points and are not interior points.

Properties of open sets include:

  • The empty set is open. By definition, , so is open.
  • The open ball is open (hence the name).
  • For any set , is open, for .
  • The finite intersection of open sets is open.
  • The arbitrary union of open sets is open.

Once again, we will not prove these properties here, but they are not too hard to show. Just use an -ball argument.

Closed Sets​

To discuss closed sets, we need to define the concept of a limit point.

In calculus, we learned about sequences and their limits. For a sequence , we say that it converges to a limit if for any , there exists an integer such that for all , . In other words, as gets larger and larger, the terms of the sequence get closer and closer to , and any finite number of terms can be made to be within any small distance of .

Let's generalize this idea to metric spaces. For a sequence in a metric space , we say that it converges to a limit if for any , there exists an integer such that for all , .

When we consider the condition , we can think of it as saying that the points are eventually contained within the open ball . As such, we can also think about convergence in terms of open balls:

Definition (convergence) A sequence in a metric space converges to a limit if for any open ball , there exists an integer such that for all , .

Topological Spaces​

In abstract algebra, we learn about fields to genearlize the real numbers. Likewise, in topology, we learn about topological spaces to generalize metric spaces.

We saw that in a space, if we have a metric, we can define open sets in that space. Now, we can think about doing the reverse, and starting with a definition of open sets, and then defining structures on the space based on that.

In other words, if we have a set , we can define a collection of subsets of that we will call "open sets". This collection is known as a topology on . Obviously, since they are called "open sets", we want them to satisfy the properties of open sets that we saw earlier:

  • The empty set and the entire set are in the topology.
  • The finite intersection of sets in the topology is also in the topology.
  • The arbitrary union of sets in the topology is also in the topology.

This is great; even without a metric, we can define various structures, such as continuity, convergence, and connectedness, using only the concept of open sets.

If is a set and is a topology on , then the pair is called a topological space. Let's consider some concrete examples.