On Monday, we investigated the appearance of points at infinity in projective geometry. Today, we turn our attention to a different field of mathematics: topology.
Topology is a favorite field among writers of popular mathematics. It is often described as rubber-sheet geometry, a field in which a donut is the same thing as a coffee mug, a field in which mathematicians think about such fanciful objects as the Möbius strip or the Klein bottle. This is fine, but here we’re going to look at the basics of topology from a more down-to-earth vantage point.
Topology is the study, unsurprisingly, of topological spaces. And what is a topological space? It is a set together with a set, often called , of distinguished subsets of satisfying certain rules. is then called a topology on , and the elements of are called open sets. The rules that such a pair must satisfy in order to earn the name topological space are as follows.
- and .
- Any union of (possibly infinitely many) elements of is in .
- If and are in , then their intersection, , is in .
To illustrate this definition, we consider two examples that you are already familiar with, whether or not you know it. First, let be any set, and let consist of all subsets of , i.e. . easily satisfies the three axioms listed above and is known as the discrete topology on .
Second, consider the set of real numbers, . You probably learned in elementary school that, if are real numbers, then is the set of all real numbers such that , and is called the open interval between and . We can also consider open intervals of the form , , and , defined in the obvious way. We can also say that, if is a real number, then . If we now let consist of all subsets of that can be written as unions of open intervals, then is a topological space, and is often referred to as the standard or order topology on .
The axioms for topological spaces are very broad and capture a great variety of different structures. This is of course appealing, but it also makes it difficult to prove interesting theorems about arbitrary topological spaces. Therefore, topologists typically consider topological spaces with certain additional properties. One of the most important of these properties, which gradually emerged in the 19th century through the investigation of infinite sequences and continuous functions of real numbers, is compactness.
Very vaguely speaking, a topological space is compact if certain phenomena in the space can be captured by a finite amount of information. But let us be precise. First, if is a topological space and is a subset of (i.e. is a collection of open sets), then we say is an open cover of if the union of the elements of is all of , i.e. for all , there is such that . We say that is compact if, whenever is an open cover of the space, then there is a finite subset of that is also an open cover. Put more elegantly, is compact if every open cover of has a finite subcover.
Let us examine our two examples of topological spaces to determine whether or not they are compact. First, consider the discrete topology on the set of natural numbers, i.e. the topological space , where . Let be the collection of all 1-element subsets of , i.e. . Then is an open cover for our space, but it has no finite subcover. Indeed, if we remove any element from , then it no longer covers our space (for example, if we remove , then we no longer cover 3). Therefore, is not compact.
A similar phenomenon happens in , where is the standard topology on . In this case, let be the set of all open intervals , where is a natural number. Then is an open cover of , but it has no finite subcover. Indeed, if is any finite subset of , then there is some largest natural number such that . Then is not covered by .
In both of these cases, you could say that the underlying reason that our space fails to be compact is that there is a sequence of elements of the space such that no subsequence of it converges to any element of the space. The sequence “escapes” our space. It “goes to infinity.” For both of our examples, this can be illustrated by the sequence of natural numbers .
Compactness is a very appealing property of a topological space, and one can prove all sorts of nice things about compact spaces, so one might be interested in taking an arbitrary topological space that is not compact and studying it by enlarging it to make it compact. If one enlarges it too much, though, the new space might not have anything to do with , so we wouldn’t be able to learn anything about from this new space. We will speak imprecisely here and say that a compactification of a space is a topological space that is compact and extends but doesn’t extend too much. (Somewhat more precisely, we require that is dense in the new space.)
How can we compactify a space? Well, we saw that a typical reason that a space fails to be compact is that it has a sequence that escapes to infinity. Therefore, you might naively think, we should simply add points at infinity to capture these sequences. And you would be correct! (Although you would have to be a little bit more specific.)
Any space has many compactifications, but two are of particular interest. There is a largest (in some sense) compactification of called the Stone–Čech compactification. We will return to this at a later date. There is also a smallest compactification, the Alexandroff compactification, which consists of just adding a single point at infinity to the space.
Let us see how the Alexandroff compactification works on the space , where is the discrete topology on . We will add one point, which we will suggestively call , and some new open sets containing . Intuitively, we want any open set containing to also contain all sufficiently large natural numbers. More precisely, the compactification will be the space , where contains all of the sets in plus all sets of the form , where is any subset of containing all but at most finitely many natural numbers.
You can check that is indeed a topological space. Let us prove that it is compact. To do this, consider an open cover of our space. We must find a finite subcover of . We first take care of . Since covers our space, it must have an element that contains . Thus, it must have an element of the form , where contains all but finitely many natural numbers. Now, for each natural number that is not in , is also covered by , so we can choose a set such that . Now consider . is a finite subset of , and we claim that it still covers the entire space. Indeed, and every natural number in is covered by , and every natural number not in is covered by . We have thus proven that is indeed compact!
We will not explicitly describe the Alexandroff compactification of the standard topology on the real numbers, but we mention in passing that it has an appealing intuitive description: it is equivalent (homeomorphic, to be precise) to the standard topology on the circle! Essentially, the space is compactified by taking the two ‘ends’ of the real line and using a new point (called , of course), to ‘stick’ them together, thus creating a circle. Similarly, the Alexandroff compactification of the real Euclidean plane is the sphere; the identification of these two spaces involves the stereographic projection, a function that finds application not just in mathematics but in fields such as cartography, crystallography, geology, and photography (in certain fish-eye lenses).
A similar construction will work on an arbitrary topological space , and this illustrates again the conceptual power of infinity: any topological space, no matter how pathologically incompact it may be, can be made compact by adding a single point at infinity!