Topological Spaces — A Primer by Jeremy Kun.
From the post:
In our last primer we looked at a number of interesting examples of metric spaces, that is, spaces in which we can compute distance in a reasonable way. Our goal for this post is to relax this assumption. That is, we want to study the geometric structure of space without the ability to define distance. That is not to say that some notion of distance necessarily exists under the surface somewhere, but rather that we include a whole new class of spaces for which no notion of distance makes sense. Indeed, even when there is a reasonable notion of a metric, we’ll still want to blur the lines as to what kinds of things we consider “the same.”
The reader might wonder how we can say anything about space if we can’t compute distances between things. Indeed, how could it even really be “space” as we know it? The short answer is: the reader shouldn’t think of a topological space as a space in the classical sense. While we will draw pictures and say some very geometric things about topological spaces, the words we use are only inspired by their classical analogues. In fact the general topological space will be a much wilder beast, with properties ranging from absolute complacency to rampant hooliganism. Even so, topological spaces can spring out of every mathematical cranny. They bring at least a loose structure to all sorts of problems, and so studying them is of vast importance.
Just before we continue, we should give a short list of how topological spaces are applied to the real world. In particular, this author is preparing a series of posts dedicated to the topological study of data. That is, we want to study the loose structure of data potentially embedded in a very high-dimensional metric space. But in studying it from a topological perspective, we aim to eliminate the dependence on specific metrics and parameters (which can be awfully constricting, and even impertinent to the overall structure of the data). In addition, topology has been used to study graphics, image analysis and 3D modelling, networks, semantics, protein folding, solving systems of polynomial equations, and loads of topics in physics.
Topology offers an alternative to the fiction of metric distances between the semantics of words. It is a useful fiction, but a fiction none the less.
Deep sledding but well worth the time.