The Unapologetic Mathematician

Mathematics for the interested outsider

Continuous Maps

Okay, we know what a topological space is. As we might expect, these will be the objects of some category \mathbf{Top}. So we need morphisms to connect them, and for this we will use the concept of a continuous map. In some sense these will be “closeness preserving”, but as we’ll see they work a little differently than the various algebraic categories we’re used to.

Here’s the definition: a continuous map from the topological space (X,\tau_X) to the topological space (Y,\tau_Y) is a function f:X\rightarrow Y between the underlying sets so that the preimage of an open set is open. That is, for every open set U\in\tau_Y we can construct f^{-1}(U)=\{x\in X|f(x)\in U\}, and we require that this set be open in X. That is: f^{-1}(U)\in\tau_X. We have a special term for isomorphisms in this category: “homeomorphism”. That is, a homeomorphism between topological spaces is a continuous map with a continuous inverse.

This always feels a little weird to me. Over in algebra we take sets, add extra structure, and define the morphisms to be those functions which preserve that structure. Here we’re defining the morphisms to be those which reflect the extra structure. But let’s accept it and move on.

Of course we also want to consider the categorical perspective. What does a continuous map give us in terms of the topology? It’s a functor from the topology on Y to the topology on X, considered as categories! Indeed, if U\subseteq V are two open sets in Y, then f^{-1}(U)\subseteq f^{-1}(V) — everything that lands in U also lands in V — and so we send open sets to open sets and inclusion arrows to inclusion arrows. This shows that f^{-1}:\tau_Y\rightarrow\tau_X is a functor.

Even more is true. First, notice that everything in X goes to some point of Y, so f^{-1}(Y)=X. Similarly, nothing lands in the empty subset of Y, so f^{-1}(\varnothing)=\varnothing. Given a family of open sets U_\alpha\in\tau_Y, a point that lands in their union is in the preimage of at least one of the sets — f^{-1}(\bigcup\limits_\alpha U_\alpha)=\bigcup\limits_\alpha f^{-1}(U_\alpha). Similarly we see that given a finite collection of open sets U_i\in\tau_Y, a point that lands in their intersection is in the preimage of each of them — f^{-1}(\bigcap\limits_{i=1}^nU_i)=\bigcap\limits_{i=1}^nf^{-1}(U_i). That is, f^{-1} preserves the finite products and arbitrary coproducts that we assume to exist in these categories.

The catch here is that, as far as we’ve come, we can’t really recover a function f from this functor f^{-1}. That is, f^{-1} only talks about open sets rather than about points of our spaces. As yet, these two definitions aren’t quite equivalent, and we’re stuck with talking about the map f as fundamental and deriving from it the functor f^{-1}.

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November 12, 2007 - Posted by | Point-Set Topology, Topology

12 Comments »

  1. “This always feels a little weird to me. Over in algebra we take sets, add extra structure, and define the morphisms to be those functions which preserve that structure.”

    In topology the morphisms also preserve the extra structure, you said it yourself: a continuous function is closeness preserving!

    To make it precise consider the relation between points and subsets of a topological space given by the point lying in the closure of the subset. Continuous functions are exactly those that preserve the relation. In other words, a function is continuous if and only if for any subset of the domain, the image of its closure is contained in the closure of its image.

    Comment by Omar | November 12, 2007 | Reply

  2. Yes, that’s true. But we haven’t defined “closeness”. What we defined is the collection of open (dually, closed) sets, and that additional structure is reflected, not preserved.

    Now, what you’re proposing is close to what we’ll end up doing. You’re proposing defining a topology by a relation between the points of a space and its category of subsets, which ultimately boils down (I think) to the Kuratowski closure axioms. But as yet that’s not what we’ve defined, and it’s not what most introductions to topology define.

    The upshot is that we can (and will!) make topology feel more algebraic, but the usual roads in feel sort of backwards.

    Comment by John Armstrong | November 12, 2007 | Reply

  3. I wonder why more textbooks don’t go the route of the closure axioms.

    Comment by nbornak | November 12, 2007 | Reply

  4. More textbooks don’t make use of the closure axioms because they are not that useful to most users of point-set topology. Most spaces that show up in nature are metric spaces, and there the metric is the natural thing to consider. The non-metric spaces which do show up (say, spaces of leaves of foliations) are most naturally thought of using open sets, etc. Even things like Frechet spaces (which have metrics, though the metric is not all that natural or useful) are best though of in terms of open sets.

    In general, point-set topology is a dead area whose foundations should be developed as efficiently as possible, without undue worries about philosophical issues…I can’t think of any open questions in the subject whose solution would have serious ramifications elsewhere.

    Comment by Andy P. | November 12, 2007 | Reply

  5. I agree with you up to a point, Andy. But everything old is constantly new again.

    There are often many equivalent ways to say the same thing, but “equivalent” doesn’t mean “equal”. While the open-set definition might be particularly efficient at proving things we used to want to know, maybe going forward we’ll need other techniques which will generalize in different ways. Having a wide view on the foundations of a subject frees you to use different approaches as are warranted. When Crans worked out the Lie 2-algebra corresponding to a Lie 2-group she took a path through quandles that someone who only ever knew the most common construction would never have dreamed of, and may not have even considered that an alternate construction would ever be useful.

    Besides questions of utility, there’s something to be said for the elegance of alternative viewpoints. Slice it one way and it looks like calculus. Slice it another way and it looks like algebra. Maybe it’s not useful for solving this problem or that, but I think it’s pretty cool.

    Oh, and I think many algebraic geometers would disagree with you on that “most spaces that show up in nature are metric spaces”.

    Comment by John Armstrong | November 12, 2007 | Reply

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  11. The dash in “we require that this set be open in X — f^{-1}(U)\in\tau_X” is confusing, I thought for a few seconds that there was a set difference there.

    Comment by Rafael | February 22, 2011 | Reply

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