Well, I’m not quite done with the updates of the topics, but I’ve gotten a number of other things done on my break. Now there’s a search bar over on the right, and the WordPress bug for subtopics has been handled. Rather than delay any longer, I guess I should jump back into the thick of it.
Topology is, roughly speaking, the study of spaces where we have an idea of what it means for points to be “close” to each other, and functions which “preserve closeness”. We don’t care about anything but the most general notion of shape. There’s the famous example of a coffee mug and a doughnut being “the same” to a topologist because they both have one hole, and if you make them out of clay you can deform one into the other without making any drastic changes like a sharp cut. In fact, it’s common to say that topology is all about situations like this, where our shapes are made from clay or rubber sheets that can be deformed around, but as we’ll see there are plenty of situations where we can make cuts (as long as we sew them up again nicely) or even weirder things can happen. Deformations are a good intuition for some aspects of topology, but they’re definitely not the most general.
Okay, so how can we get a handle on this notion of “closeness”. The usual way is to take the set of points we’re looking at and define some collection of its subsets as the “open” subsets. Such a collection is required to satisfy a few rules:
- The empty set and the whole set are both in
- The union of any collection of subsets in is again in
- The intersection of any finite collection of subsets in is again in
We call the specified collection a “topology” on the set , and pair of a set and a topology on we call a “topological space. The elements of we call the open sets of , and their complements in we call the closed sets.
Notice here that the collection of closed sets is completely determined by the collection of open sets. This leads to an alternate viewpoint, where we define a collection of subsets of satisfying:
- The empty set and the whole set are both in
- The intersection of any collection of subsets in is again in
- The union of any finite collection of subsets in is again in
Now the elements of are called the closed subsets of the topological space, and their complements are called the open subsets.
We can put more than one topology on the same set , and we can compare different topologies. Let’s say that we have topologies and on a set , so that . That is, every subset of that calls open, does as well. In this case, we say that the topology is “coarser” than , or that is “finer” than . Since we define this relationship by restricting subset containment from to those collections of subsets of which are actually topologies, it defines a partial order on the collection of all topologies on .
The coarsest possible topology is , which says that only the empty subset and the whole set are open. We call this the “trivial” or the “indiscrete” topology on . Conversely, the finest possible topology is , which says that every subset is open. This we call the “discrete” topology on . Useful topologies tend to fall somewhere between these two extremes, but at least we know that has a top and a bottom element for the coarseness relation.
In the middle, let’s say we have some collection of topologies. Then we can define their intersection as subsets of . This will also be a topology, as is easily shown from the definition above. It is the finest topology which is coarser than all the topologies in , and so any subset of has a greatest lower bound.
On the other hand, the union of this collection may not be a topology, which could serve as a least upper bound. However, there is always at least one topology that contains this union — the discrete topology. So we can consider the collection — known to be nonempty — of all topologies which contain the union . The intersection of this collection of topologies will be a topology (as above) which is finer than each topology , and is the coarsest possible such topology. Thus any subset of has a least upper bound.
Together, these results say that the is a complete lattice under the coarseness relation. This turns out to be useful when we have some set we want to put a topology on, and we want to do it in the coarsest possible way subject to a collection of requirements. The fact that is a complete lattice says that we can find the coarsest possible topology satisfying the relations one at a time, and then we can find the coarsest topology finer than each of them.