The Unapologetic Mathematician

Mathematics for the interested outsider

Topology

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 X we’re looking at and define some collection T\subseteq P(X) of its subsets as the “open” subsets. Such a collection is required to satisfy a few rules:

  • The empty set \varnothing and the whole set X are both in T
  • The union \bigcup\limits_\alpha U_\alpha of any collection \{U_\alpha\}\subseteq T of subsets in T is again in T
  • The intersection \bigcap\limits_{i=1}^nU_i of any finite collection \{U_i\}\subseteq T of subsets in T is again in T

We call the specified collection T a “topology” on the set X, and pair of a set X and a topology T on X we call a “topological space. The elements of T we call the open sets of X, and their complements in X 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 T\subseteq P(X) of subsets of X satisfying:

  • The empty set \varnothing and the whole set X are both in T
  • The intersection \bigcap\limits_\alpha U_\alpha of any collection \{U_\alpha\}\subseteq T of subsets in T is again in T
  • The union \bigcup\limits_{i=1}^nU_i of any finite collection \{U_i\}\subseteq T of subsets in T is again in T

Now the elements of T 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 X, and we can compare different topologies. Let’s say that we have topologies T_1 and T_2 on a set X, so that T_1\subseteq T_2. That is, every subset of X that T_1 calls open, T_2 does as well. In this case, we say that the topology T_1 is “coarser” than T_2, or that T_2 is “finer” than T_1. Since we define this relationship by restricting subset containment from P(P(X)) to those collections of subsets of X which are actually topologies, it defines a partial order on the collection \mathcal{T} of all topologies on X.

The coarsest possible topology is \{\varnothing,X\}, which says that only the empty subset and the whole set are open. We call this the “trivial” or the “indiscrete” topology on X. Conversely, the finest possible topology is P(X), which says that every subset is open. This we call the “discrete” topology on X. Useful topologies tend to fall somewhere between these two extremes, but at least we know that \mathcal{T} has a top and a bottom element for the coarseness relation.

In the middle, let’s say we have some collection \{T_\alpha\}\subset\mathcal{T} of topologies. Then we can define their intersection \bigcap\limits_\alpha T_\alpha as subsets of P(X). 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 \{T_\alpha\}, and so any subset of \mathcal{T} 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 \bigcup\limits_\alpha T_\alpha. The intersection of this collection of topologies will be a topology (as above) which is finer than each topology T_\alpha, and is the coarsest possible such topology. Thus any subset of \mathcal{T} has a least upper bound.

Together, these results say that the \mathcal{T} 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 \mathcal{T} 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.

November 5, 2007 Posted by | Point-Set Topology, Topology | 18 Comments

   

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