Metric Spaces and Topology in Analysis

[Mr Davis 97]

I'm beginning to study analysis beyond real numbers, but I am a but confused. What is the relation between topology, metric spaces, and analysis? From what it seems, it's that metric space theory forms a subset of topology, and that analysis uses the metric space notion of distance to describe limiting processes. Is this at all correct? What exactly distinguishes topology and analysis? Are there limits in topology? It seems that there are things called accumulation points in topology, and these seem similar to limits in analysis, but all seems jumbled together and a bit confusing. Here's another question: Why do we tend to learn analysis and then topology, rather than topology and then analysis? It seems that topology is a prerequisite to analysis.

Basically, what is the clear delineation between topology and analysis, and what is the relation between the two?

 

[fresh_42]

I'm beginning to study analysis beyond real numbers, but I am a but confused. What is the relation between topology, metric spaces, and analysis? From what it seems, it's that metric space theory forms a subset of topology, and that analysis uses the metric space notion of distance to describe limiting processes. Is this at all correct?

More or less. Topology is about spaces with open and closed sets and continuous functions. Now if we have a metric, we can define such a topology by the metric, so metric spaces are special topological spaces. Analysis deals with $\mathbb{R}^n$ and $\mathbb{C}^n$ and the functions there. As both, $\mathbb{R}^n$ and $\mathbb{C}^n$, are metric spaces, analysis is a certain example of a topological space. Yes, we use distances and norms in analysis, given by the Euclidean norm. However, other measures are used, too, when it comes to integration. You can look up Lebesgue measure, Borel measure and $\sigma-$algebras. Analysis is simply more than some continuous functions on open intervals.

What exactly distinguishes topology and analysis? Are there limits in topology?

As you already mentioned, the metric distinguishes them, resp. specifies analytic spaces. Yes, there are limits in topology; usually in metric topologies. But there is a generalization to other spaces, too. Look up nets. Another kind of limits are the points of a closed set without its interior: the boundary.

It seems that there are things called accumulation points in topology, and these seem similar to limits in analysis, but all seems jumbled together and a bit confusing.

Accumulation points or limit points are usually used if a metric is given, since you need something to tell what "closer than" means. In general, topology is a complete different field than analysis. Analysis plays in a theater, which also can be found in the "Topological Guide to Theaters", but not the other way around. A metric is a strong tool which general topological spaces do not have.

Here's another question: Why do we tend to learn analysis and then topology, rather than topology and then analysis? It seems that topology is a prerequisite to analysis.

You can see it this way, and an answer to a why question is always an opinion. Mine is, that analysis is closer to what we called mathematics at school, so it is a natural way to expand this way. It is also more useful for any sciences, regardless whether STEM related or not. You don't need to know what a covering is as an economist, but you should know what a differentiation is. We also start to learn integers and not groups and rings, rational and real numbers and not fields; we learn to prove statements without a course in logic. In this sense it is easier to say: the integers are an example of a commutative ring, and, an open interval is an example of an open set, than to learn an entire building - btw. a topological term - with many unnecessary floors. Sure you can start with a half group, learn group theory, ring theory, field theory and Galois theory only to introduce counting:
$$\mathbb{N}\longrightarrow \mathbb{N}_0\longrightarrow \mathbb{Z}\longrightarrow \mathbb{Q}\longrightarrow \mathbb{R}\longrightarrow \mathbb{C}$$
The same holds for topology. You can learn what a refinement is, but you probably won't need it in analysis.

Basically, what is the clear delineation between topology and analysis, and what is the relation between the two?

In topology we have the concept of open sets and continuous functions. That's it. In analysis you have entire families of functions which are investigated. However, we are stick to real or complex numbers, resp. vector spaces. Neither exists in a general topological space.

 

[math771]

It's great that you're starting to delve into analysis beyond real numbers! It can definitely be a bit confusing at first, but understanding the relationship between topology, metric spaces, and analysis can help clarify things.

First of all, you are correct in thinking that metric space theory is a subset of topology. In fact, metric spaces are a specific type of topological space, where the topology is induced by a metric (i.e. a way to measure distance between points). This means that the concepts and techniques used in topology can also be applied to metric spaces.

Now, moving on to analysis, it is a branch of mathematics that deals with the study of continuous and differentiable functions, as well as their properties. In order to understand these functions, we often use the concept of limits, which is where metric spaces come into play. As you mentioned, the metric space notion of distance is used to describe limiting processes, such as finding the limit of a sequence of numbers or the limit of a function as it approaches a certain point.

So, what distinguishes topology and analysis? While both deal with the study of spaces and their properties, topology focuses on the "shape" or structure of a space, while analysis delves into the behavior of functions on that space. In other words, topology is concerned with the overall properties of a space, while analysis looks at the specific functions and how they behave on that space.

As for your question about learning analysis before topology, it's true that many people tend to learn analysis first. This is because topology is often seen as a prerequisite for analysis, as it provides the necessary tools and concepts for understanding the behavior of functions. However, some people do choose to learn topology first, so it really depends on personal preference and the specific curriculum of your courses.

In terms of limits in topology, there are indeed concepts similar to limits in analysis, such as accumulation points and limit points. These are points that are "close" to a given set in a topological space, but may not necessarily be part of the set itself. This can be thought of as a generalization of limits in analysis, where we are concerned with points that a function approaches, rather than just specific numbers.

Overall, the relationship between topology and analysis is that topology provides the framework and tools for understanding the structure of a space, while analysis uses these concepts to study the behavior of functions on that space. I hope this helps clarify things for you!