A collection of 14 posts
In this post, I will prove a series of seemingly random conjectures about compactness. However, each of them will be required in order to prove the main result of this post: The Heine-Borel Theorem. This theorem completely characterizes compact sets in euclidean space.
Unlike the other topological invariants we've discussed, such as connectedness, compactness does not have an immediately obvious geometric description. However, it is a concept that is impossible to avoid, as it shows up everywhere and in every field of mathematics.
Now we can think about a different type of "connectedness." Intuitively, if a space is "connected" you should be able to draw a path between any two points in the space. Otherwise, if there are points in the space that cannot be connected by a path, it is "disconnected."
In this post, I'm going to prove the Intermediate Value Theorem and the One-Dimensional Brouwer Fixed Point Theorem, which are two results that are undeniably and unreasonably useful. In order to prove them, however, we will need to study the notion of connectedness.
I'm now going to talk about sequences and nets, which often provide an alternative way of describing topological phenomena. I'll also talk about Hausdorff spaces, which have all sorts of nice properties.
The notion of a quotient space will effectively allow us to glue pieces of topological spaces together. This corresponds to the collapsing of equivalent subsets to points which occurs in quotient sets, as I mentioned in my last post.
Next let's talk about an intuitive way to combine topological spaces to create new spaces which inherit certain characteristics from their parents. We've talked about Cartesian products before in the context of set theory, but what happens if we take the Cartesian product of topological spaces?
A topological space is, at its core, just a set with some additional structure. So what if we want to keep the structure, but change the underlying set? There's an easy and somewhat obvious way to do this.
The concept of continuity is central to the study of topology. So much so, in fact, that whenever anyone talks about a map between topological spaces, they generally expect you to know that they're talking about a continuous map.
It's fairly common to think of open sets as sets which do not contain their boundary, and closed sets as sets which do contain their boundary. The trouble here lies in defining the word 'boundary.'
Essentially, a basis is a 'small' collection of open sets from which every open set can be easily generated. It is often useful to talk about the topology generated by a specific basis, since many facts about a topology can be gleaned by studying one of its bases.
There are also many circumstances in which we care about the shape of a space but couldn't care less about distances. For instance, a famous puzzle that influenced the development of the entire field of topology is the problem of the **Seven Bridges of Königsberg**.