Interiors and Closures

When we pick a topology $\tau$ on a set $X$ , not every subset is open, and not every subset is closed. However, we can still come up with some open and closed sets from any subset $U$ .

For the open set, notice that we always have at least one open set inside $U$ : the empty set. So we can gather up all the open sets contained in $U$ and take their union. Since they’re all contained in $U$ the union will be as well. And since arbitrary unions of open sets are still open, it’s an open set. In fact, it’s the largest open set contained in $U$ , because it contains all the other open subsets of $U$ . We call this the “interior” of $U$ , denoted $\mathrm{int}(U)$ or $U^\circ$ . Clearly the interior of an open set it the set itself.

Dually, we know that there is at least one closed set containing $U$ : the whole space $X$ . Then the intersection of all the closed sets containing $U$ will be a closed set containing $U$ , and will be the smallest such closed set. We call this the “closure” of $U$ and write $\mathrm{Cl}(U)$ or $\overline{U}$ . As for the interior, the closure of a closed set is the set itself.

Now the complement of the closure of $U$ is an open set contained in the complement of $U$ . In fact, any other open set contained in the complement of $U$ will be contained in this one, so it is the interior of the complement of $U$ . Dually, the closure of the complement of $U$ is the complement of the interior of $U$ .

We can write this fact down categorically as well. Since it reverses subset containment, complementation is a contravariant equivalence from the poset $P(X)$ of subsets of $X$ (considered as a category) to itself. That is, $P(X)$ is equivalent to $P(X)^\mathrm{op}$ . The interior and closure operators are covariant functors from $P(X)$ to itself, since they preserve containment. The previous paragraph states that these two functors are dual to each other, in the sense that $\mathrm{Cl}^\mathrm{op}:P(X)^\mathrm{op}\rightarrow P(X)^\mathrm{op}$ is the same functor as $\mathrm{int}:P(X)\rightarrow P(X)$ under the above equivalence. So all the really important information is contained in the closure functor.

Now, what do we know about this functor? Well, since $U$ is contained in $\mathrm{Cl}(U)$ we have a natural transformation $\eta:1_{P(X)}\rightarrow\mathrm{Cl}$ . Then since $\mathrm{Cl}(\mathrm{Cl}(U))$ is contained in $\mathrm{Cl}(U)$ we have a natural transformation $\mu:\mathrm{Cl}^2\rightarrow\mathrm{Cl}$ . I haven’t really covered these yet, but it’s straightforward from here to verify that $\mathrm{Cl}$ along with these two natural transformations forms a monad. If you’re interested in learning more right away, go check out The Catsters’ series of YouTube videos.

We also can easily check that $\mathrm{Cl}(U\cup V)=\mathrm{Cl}(U)\cup\mathrm{Cl}(V)$ , and that $\mathrm{Cl}(\varnothing)=\varnothing$ . That is, the functor $\mathrm{Cl}$ preserves all finite coproducts. It turns out that this is enough to characterize the topology in its entirety!

Given a set $X$ , a closure operator on $X$ is a monad $(\mathrm{Cl},\eta,\mu)$ , where $\mathrm{Cl}:P(X)\rightarrow P(X)$ is a functor which preserves finite coproducts. This data is equivalent to the four axioms given by Kuratowski:

$U\subseteq\mathrm{Cl}(U)$
$\mathrm{Cl}(\mathrm{Cl}(U))=\mathrm{Cl}(U)$
$\mathrm{Cl}(U\cup V)=\mathrm{Cl}(U)\cup\mathrm{Cl}(V)$
$\mathrm{Cl}(\varnothing)=\varnothing$

From here we can define the closed sets of $X$ to be those in the image of the functor $\mathrm{Cl}$ . From axiom 1 we see that $X\subseteq\mathrm{Cl}(X)$ , but this closure must be a subset of $X$ , and so $X$ is closed. Axiom 4 tells us straight off that $\varnothing$ is closed. Axiom 3 tells us that the finite union of closed sets is closed. We just need to know that arbitrary intersections of closed sets are again closed.

For this, we note that given any collection $\{U_\alpha\}$ of subsets, the intersection $\bigcap\limits_\alpha U_\alpha$ lies in each $U_\alpha$ , and so by functoriality $\mathrm{Cl}(\bigcap\limits_\alpha U_\alpha)\subseteq\mathrm{Cl}(U_\alpha)$ for each $\alpha$ . Thus we see that $\mathrm{Cl}(\bigcap\limits_\alpha U_\alpha)\subseteq\bigcap\limits_\alpha\mathrm{Cl}(U_\alpha)$ . In particular, if all the $U_\alpha$ are closed, then $\mathrm{Cl}(\bigcap\limits_\alpha U_\alpha)\subseteq\bigcap\limits_\alpha U_\alpha$ . But since $U\subseteq\mathrm{Cl}(U)$ for all $U$ , this inclusion is actually an equality, and thus the intersection of the $U_\alpha$ is in the image of the closure functor. And thus we really have constructed the closed sets of a topology on $X$ .

November 13, 2007 - Posted by John Armstrong | Point-Set Topology, Topology

11 Comments »

Topologies are a type of monad? How unexpected.

Comment by nbornak | November 13, 2007 | Reply
Actually, just about closure operator can be construed as a monad acting on a poset, typically a lattice. For example, let G be a group, and consider the operator Gr: P(G) –> P(G) which assigns to each subset U of G the subgroup Gr(U) it generates (i.e., close up U under the group operations of G). Then Gr is a monad acting on the poset P(G). Examples of this sort are easily multiplied. It even seems reasonable to define a closure operator as a monad acting on a poset.

But it’s a pleasant fact that, as John said, topologies on X are precisely tantamount to monads P(X) –> P(X) that preserve finite unions (= coproducts). (Or, alternatively, to comonads which preserve finite intersections.)

Other monads which don’t act on posets but which still behave like closure operators are: completion of a metric space (as a monad on the category of metric spaces and uniformly continuous maps), and completion of an integral domain to its field of quotients. In cases like these, the “completion of the completion is isomorphic to the completion”; more precisely, the multiplication $\mu: MM \to M$ is an isomorphism. Such monads are called idempotent.

Comment by Todd Trimble | November 13, 2007 | Reply
That seriously rocks. Thanks for the insight.

Comment by nbornak | November 13, 2007 | Reply
S’what I’m here for. With ample assistance from Todd, of course.

Actually, to be honest a certain amount of the categorical interpretations are things I’ve idly thought should be there, but never worked out the details before. Presenting point-set topology here gives me the excuse to work it out as I go. The upshot is that there may be even cooler ways to say it, but I don’t notice them at a first (well, second) pass.

Comment by John Armstrong | November 13, 2007 | Reply
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