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.