Whence topology: main concepts
Having introduced you to the swamp and the monsters, let me finish by sketching the structures used to make some sense out of it all.
Spaces
Very briefly, a (topological) space is a set endowed with a notion of “neighborhood”. Examples include
- geometric objects,
- families of geometric objects, and
- families of functions.
The word “space” in this context has nothing whatever to do with three degrees of freedom.1 It just means that this set is being regarded as a sort of geometric object in itself, and that the properties of it that we are interested in come from the given notion of “neighborhood.”
The conventions on what constitute an acceptable notion of “neighborhood” are very permissive. Here they are, with “neighborhood” shortened to “nbd”:
- Nbds of a point contain that point. That is, if \(N\) is a nbd of \(x\), then \(x \in N\).
- Supersets of nbds are nbds too. That is, if \(N\) is a nbd of \(x\) and \(N \subset U\), then \(U\) is a nbd of \(x\).
- Finite intersections preserve nbds. That is, if \(M\) and \(N\) are nbds of \(x\), then \(M \cap N\) is a nbd of \(x\).
- Nbds in nbds: if \(N\) is a nbd of \(x\), then there is a nbd \(M\) of \(x\) such that \(N\) is a nbd of every point in \(M\).
This is not the usual definition, but it is equivalent to it.
As an example, let’s say a subset \(N\) of the plane is a nbd of a point \(p\) when there is an open disc around \(p\) that fits inside \(N\). This endows the plane with a notion of “neighborhood” satisfying the above axioms. (The third axiom is the least easy to prove for this example.)
You can generalize this approach to sets with fairly arbitrary notions of distance. These are called metric spaces, and are among the nicest kinds of topological space.
Even so, it is not at all clear from this definition that one can do much of worth with it at all. The first indication it might be useful comes from the next main concept.
Continuity
Suppose \(X\) and \(Y\) are topological spaces. A function \(f: X \to Y\) is continuous when it “respects the notion of neighborhood.” Intuitively we want continuous functions to be those that take “nearby points to nearby points.”
There are at least two approaches to making this rigorous. One is to define a function to be continuous when it takes “infinitely nearby points to infinitely nearby points.” The problem with this is that, as far as I know, properly defining “infinitely nearby” takes more mathematical logic than I would like to put on my computational topology blog.
The more traditional approach is not as direct, but still works, and does not require a large up-front logical down payment. We imagine neighborhoods are the appropriate possible meanings for “nearby.” Then a function is continuous at \(x\) when, no matter what appropriate meaning we pick for being “nearby \(f(x)\)”, there is an appropriate meaning for “nearby \(x\)” such that \(f\) takes points “nearby \(x\)” to points “nearby \(f(x)\)”. That is, \(f\) is continuous at \(x\) when, for all neighborhoods \(M\) of \(f(x)\), there is a neighborhood \(N\) of \(x\) such that \(f(N) \subset M\). It’s continuous with no further qualifications when it’s continuous at all points in its domain.
A notion of neighborhood is thus the bare minimum amount of structure needed on sets to tell you what it means for a function from one to the other to be continuous. Topology is the study of these sets and functions.
Equivalence
Whenever defining a field of mathematical study, it is important to make clear what is meant by two mathematical objects being “basically the same” or “equivalent”. For instance, in set theory, two sets are “equivalent” when there is a one-to-one, onto function between them, a bijection between them. In order theory, two ordered sets are considered equivalent when there is an order-preserving bijection from one to the other. In group theory, two groups are isomorphic when there is a bijective multiplication-preserving map from one group to the other. In geometry, two metric spaces are isometric when there is a distance-preserving bijection from one space to the other.
In topology, the problem of equivalence is more subtle. One is tempted to say, following the previous notions, that two spaces are equivalent when there is a continuous bijection from one to the other. This doesn’t work, for a silly reason. On every set \(S\), there is a very easy topology to construct, the indiscrete topology. In the indiscrete topology on \(S\), the only nbd of any point \(x\) is the whole set \(S\). If \(Y\) is an indiscrete topological space constructed this way, then every function \(f: X \to Y\) is continuous. If we allowed “continuous bijection” as our notion of equivalence, the whole field would reduce to set theory.
But there is another perspective that does work. Notice that in set theory, equivalences are the same thing as invertible functions. The same is true in order theory, group theory, and geometry. For instance, an isometry from \(X\) to \(Y\) is a distance-preserving function \(f: X \to Y\) that has a distance-preserving inverse \(g: X \to Y\). Likewise, monotone bijections have monotone inverses, and bijective homomorphisms’ inverses are also homomorphisms. In these fields it is redundant to say the inverses must lie in the same category of functions, but in topology and other fields it is essential.
So instead, in topology, a homeomorphism from \(X\) to \(Y\) is a continuous function \(f: X \to Y\) admitting a continuous inverse \(g: Y \to X\). The existence of a homeomorphism is the right notion of equivalence, usually written \(\approx.\) On the whole, it is a fairly weak notion of equivalence. Circles, triangles, squares, and knots, just to give an example, are all homeomorphic. So notions of distance, angle, or even straight line are thrown right out of consideration.
It is hard to imagine at first that any properties are left unchanged by arbitrary homeomorphism. Yet by the very same token, a property is very fundamental indeed if not even an arbitrary homeomorphism can change it.
Connectedness
The easiest such property to prove unchangeable, once one notices it, is connectedness. For instance, the interval \(I = [0,1]\) should be connected. In general, we say that a space \(X\) is path-connected when, for any two points \(p,q \in X\), there is a path from \(p\) to \(q\)—that is, a continuous function \(\gamma: I\to X\) such that \(\gamma(0) = p\) and \(\gamma(1) = q\). Given any continuous map \(\phi: X \to Y\), if there’s a path \(\gamma: [0,1] \to X\) from \(p\) to \(q,\) then \(\phi \circ \gamma\) is a path from \(\phi(p)\) to \(\phi(q)\).
So path-connectedness is a topological invariant.
Separation
The real numbers satisfy a “betweenness” law: if \(x\) and \(y\) are distinct, then they lie in distinct open intervals. Not all spaces—not even all useful spaces—satisfy such nice properties.2 The most geometrically familiar such spaces do, though.
These properties are traditionally called separation axioms.
Compactness and convergence
This is a subtler concept. Recall that every bounded sequence of real numbers has a convergent subsequence. This is a property of real numbers, but we can take a different perspective. Picking lower and upper bounds \(L\) and \(U\), we could say instead that for every sequence \(A: \{0, 1, 2, \ldots\} \to [L,U],\) there is a convergent subsequence \(A|_S: S \to [L,U],\) where \(S\) is some infinite subset of \(\{0,1,2,\ldots\}.\) That is, every sequence in \([L, U]\) has a convergent subsequence. This looks more like a property of intervals of the form \([L,U]\) than a property of real numbers. This is basically what compactness is: compact spaces are those in which one can expect limiting processes to converge upon refinement.
Compact spaces and their relatives are the simplest topological spaces. There is a gradation of potential horribleness for spaces; compact spaces are near the nice extreme. If spaces were pets, compact spaces would be goldfish or guinea pigs.
Concluding impressions
For analysts
Topology is a necessity for analysis. It is useful for instilling doubt in the seemingly obvious, in “playing for safety” in making claims, and as a way to export geometric intuition to objects that cannot be visualized. Compactness is very useful—for instance, in stating the Arzelà-Ascoli theorem.
For geometers
Topology is the groundwork and language necessary for all the constructions you actually want to do—vector fields, Riemannian metrics, flows, etc. Compactness is very useful—for instance, in proving the Hopf-Rinow theorem.
But most of all…
Topology is the study of invariants of spaces under continuous deformation—the most fundamental properties of a geometric object. Compactness is—well, you get the picture.