# The 14 closure operations

I’ve been getting complaints that my lemmas have not been so lovely (or little) lately, so let’s do something a bit more down to earth. This is a story I learned from the book Counterexamples in topology by Steen and Seebach [SS].

A topological space comes equipped with various operations on its power set . For instance, there are the maps (interior), (closure), and (complement). These interact with each other in nontrivial ways; for instance .

Consider the monoid generated by the symbols (interior), (closure), and (complement), where two words in , , and are identified if they induce the same action on subsets of an arbitrary topological space .

Lemma. The monoid has 14 elements, and is the monoid given by generators and relations

Proof. The relations , , and are clear, and conjugating the first by shows that is already implied by these. Note also that and are monotone, and for all . A straightforward induction shows that if and are words in and , then . We conclude that

so (this is the well-known fact that is a regular open set). Conjugating by also gives the relation (saying that is a regular closed set).

Thus, in any reduced word in , no two consecutive letters agree because of the relations , , and . Moreover, we may assume all occurrences of are at the start of the word, using and . In particular, there is at most one in the word, and removing that if necessary gives a word containing only and . But reduced words in and have length at most 3, as the letters have to alternate and no string or can occur. We conclude that is covered by the 14 elements

To show that all 14 differ, one has to construct, for any in the list above, a set in some topological space such that . In fact we will construct a single set in some topological space where all 14 sets differ.

Call the sets for the noncomplementary sets obtained from , and their complements the complementary sets. By the arguments above, the noncomplementary sets always satisfy the following inclusions:
It suffices to show that the noncomplementary sets are pairwise distinct: this forces (otherwise ), so each noncomplementary set contains and therefore cannot agree with a complementary set.

For our counterexample, consider the 5 element poset given by

and let be the disjoint union of (the Alexandroff topology on , see the previous post) with a two-point indiscrete space . Recall that the open sets in are the upwards closed ones and the closed sets the downward closed ones. Let . Then the diagram of inclusions becomes We see that all 7 noncomplementary sets defined by are pairwise distinct.

Remark. Steen and Seebach [SS, Example 32(9)] give a different example where all 14 differ, namely a suitable subset . That is probably a more familiar type of example than the one I gave above.

On the other hand, my example is minimal: the diagram of inclusions above shows that for all inclusions to be strict, the space needs at least 6 points. We claim that 6 is not possible either:

Lemma. Let be a finite topological space, and any subset. Then and .

But in a 6-element counterexample , the diagram of inclusions shows that any point occurs as the difference for some . Since each of and is either open or closed, we see that the naive constructible topology on is discrete, so is by the remark from the previous post. So our lemma shows that cannot be a counterexample.

Proof of Lemma. We will show that ; the reverse implication was already shown, and the result for follows by replacing with its complement.

In the previous post, we saw that is the Alexandroff topology on some finite poset . If is any nonempty subset, it contains a maximal element , and since is a poset this means for all . (In a preorder, you would only get .)

The closure of a subset is the lower set , and the interior is the upper set . So we need to show that if and , then .

By definition of , we get , i.e. there exists with . Choose a maximal with this property; then we claim that is a maximal element in . Indeed, if , then so , meaning that there exists with . Then and , which by definition of means . Thus is maximal in , hence since and . From we conclude that , finishing the proof.

The lemma fails for non- spaces, as we saw in the example above. More succinctly, if is indiscrete and , then and . The problem is that is maximal, but and .

References.

[SS] L.A. Steen, J.A. Seebach, Counterexamples in Topology. Reprint of the second (1978) edition. Dover Publications, Inc., Mineola, NY, 1995.

# Finite topological spaces

One of my favourite bits of point set topology is messing around with easy topological spaces. What could be easier than finite topological spaces? The main result (below) is that the category of finite topological spaces is equivalent to the category of finite preorders.

Recall (e.g. from algebraic geometry) the following definition:

Definition. Let be a topological space. Then the specialisation preorder on (the underlying set of) is the relation if and only if .

Note that it is indeed a preorder: clearly , and if and , then , so , showing . We denote this preorder by .

Note that the relation is usually denoted in algebraic geometry, which is pronounced “ specialises to ”.

Definition. Given a preorder , the Alexandroff topology on is the topology whose opens are the cosieves, i.e. the upwards closed sets (meaning and implies ).

To see that this defines a topology, note that an arbitrary (possibly empty) union or intersection of cosieves is a cosieve. A subbase for the topology is given by the principal cosieves for any . We denote the set with its Alexandroff topology by .

Likewise, the closed sets in are the sieves (or downwards closed sets); for instance the principal sieves . The closure of is the sieve generated by ; for instance the closure of a singleton is the principal sieve .

Theorem. Let be the functor , and the functor .

1. Let be a preorder, a topological space, and a function. Then is a monotone function if and only if is a continuous function .
2. The functors and are adjoint: .
3. The composition is equal (not just isomorphic!) to the identity functor.
4. The restriction of to the category of finite topological spaces is equal to the identity functor.
5. If is a topological space, then is if and only if is a poset.
6. If is a preorder, then is a poset if and only if is .
7. The functors and give rise to adjoint equivalences

Proof. (1) Suppose is monotone, let be a closed subset, and let . Suppose and . Since is monotone and is closed, we get , i.e. . We conclude that , so is downward closed, hence closed in .

Conversely, suppose is continuous, and suppose in . Then , so by continuity we get , so .

(2) This is a restatement of (1): the map

is a bijection.

(3) Since , we conclude that if and only if , so the specialisation preorder on is the original preorder on .

(4) In general, the counit is a continuous map on the same underlying space, so is finer than . Conversely, suppose is closed, i.e. is a sieve for the specialisation preorder on . This means that if , then implies ; in other words . If and therefore is finite, there are finitely many such , so is the finite union

of closed subsets of . Thus any closed subset of is closed in , so the topologies agree.

(5) The relations and mean and . This is equivalent to the statement that a closed subset contains if and only if it contains . The result follows since a poset is a preorder where the first statement only happens if , and a space is a space where the second statement only happens if .

(6) Follows from (5) applied to since by (3).

(7) The equivalence follows from (3) and (4), and the equivalence then follows from (5) and (6).

Example. The Alexandroff topology on the poset is the Sierpiński space with topology . As explained in this post, continuous maps from a topological space to are in bijection with open subsets , where is sent to (and to the indicator function ).

Example. Let be a set with two elements. There are 4 possible topologies on , sitting in the following diagram (where vertical arrows indicate inclusion bottom to top):

These correspond to 4 possible preorder relations , sitting in the following diagram (where vertical arrows indicate inclusion top to bottom):

We see that finer topologies (more opens) have stronger relations (fewer inequalities).

Example. The statement in (4) is false for infinite topological spaces. For instance, if is the Zariski topology on a curve, then any set of closed points is downwards closed, but it is only closed if it’s finite. Or if is a Hausdorff space, then the specialisation preorder is just the equality relation , whose Alexandroff topology is the discrete topology.

I find the examples useful for remembering which way the adjunction goes: topological spaces generally have fewer opens than Alexandroff topologies on posets, so the continuous map should go .

Remark. On any topological space , we can define the naive constructible topology as the topology with a base given by locally closed sets for open and closed. In the Alexandroff topology, a base for this topology is given by the locally closed sets : indeed these sets are clearly naive constructible, and any set of the form for upward closed and downward closed has the property .

Thus, if is the Alexandroff topology on a preorder, we see that the naive constructible topology is discrete if and only if the preorder is a poset, i.e. if and only if is .

# Grothendieck topologies (topologies 4/6)

This post is the first goal in a series on sieves (subobjects of representable presheaves); I will give another generalisation in the next two posts. In the first post of the series, I defined sieves and gave basic examples, and last week I showed how the sheaf condition on a site can be stated in terms of sieves:

Corollary. Let be a (small) site. For a set of morphisms with the same target, write for the presheaf image of . Then a presheaf is a sheaf if and only if for every covering in , the inclusion induces an isomorphism

Thus, if is a site (a small category with a Grothendieck pretopology), we should be able to obtain the category of sheaves purely in terms of sieves. This is the notion of a Grothendieck topology that we describe at the end of this post.

Before giving the definition, note that any morphism in gives a pullback taking to its inverse image under (I avoid the word ‘pullback’ here to make sure this is truly a subpresheaf and not a presheaf with a monomorphism to defined uniquely up to unique isomorphism). Thus, is itself a presheaf (it takes values in since is small).

Also note the following method for producing sieves: if is a presheaf, a subpresheaf, and a section over some , we get a sieve by

By the Yoneda lemma, this is just the inverse image of along the morphism classifying . Note that is the maximal sieve if and only if .

Definition. Let be a small category. Then a Grothendieck topology on consists of a subpresheaf such that

1. For all , the maximal sieve is in .
2. If and with , then .
3. If is a sieve such that , then (equivalently, then is the maximal sieve ).

The sieves are called covering sieves. Since is a presheaf, we see that for any and any covering sieve , the pullback is covering. Condition 2 says that any sieve containing a covering sieve is covering. In the presence of condition 1, conditions 2 and 3 together are equivalent to the local character found in SGA IV, Exp. II, Def. 1.1:

• If with , such that for every morphism the inverse image of along is in , then .

Indeed, applying this criterion when immedately shows if , since the inverse image of along is the maximal sieve . Thus the local character implies criterion 2. The local character says that if contains a covering sieve , then is covering. Assuming criterion 2, the sieve contains a covering sieve if and only if is itself covering, so the local character is equivalent to criterion 3.

Remark. One property that follows from the axioms is that is closed under binary intersection, i.e. if then . Indeed, if for some , then

so . Axioms 2 and 3 give .

Example. Let be a pretopology on the (small) category ; see Tag 00VH for a list of axioms. For each , define the subset as those that contain a sieve of the form for some covering in . (See the corollary at the top for the definition of .) Concretely, this means that there exists a covering such that for all , i.e. is covered by morphisms that are in the given sieve .

Lemma. The association is a topology. It is the coarsest topology on for which each for is a covering sieve.

Proof. We will use the criteria of Tag 00VH. If , then there exists with . If is any morphism in , then by criterion 3 of Tag 00VH. But , because a morphism factors through if and only if factors through . Thus, , so , and is a subpresheaf of .

Condition 1 follows immediately from criterion 1 in Tag 00VH, and condition 2 is satisfied by definition. For condition 3, suppose satisfies . Then there exists with . This means that for all , i.e. for all . Thus, for each there exists in such that , i.e. for all and all . Thus, if denotes , then we get . But is a covering by criterion 2 of Tag 00VH, so .

If is any other Grothendieck topology for which each for is covering, then contains by criterion 2.

To state the obvious (hopefully), the notion of sheaf can therefore be defined on a Grothendieck topology in a way that coincides with the usual notion for a Grothendieck pretopology:

Definition. Let be a small category, and let be a Grothendieck topology. Then a presheaf is a sheaf if for any and any , the map induces an isomorphism

Thus, a Grothendieck topology is an internal characterisation (inside ) of which morphisms one needs to localise to get . In the last two posts, we will generalise this even further to a Lawvere–Tierney topology on an arbitrary topos.

# Presheaves are colimits of representables (topologies 2/6)

Last week, I started a series on sieves and Grothendieck topoi. This is a short intermezzo on a well-known lemma from category theory that I will need for next week’s instalment.

Let be a small category. Recall that a presheaf on is a functor . Examples include the representable presheaves for , given by . The Yoneda lemma says that for any presheaf and any , the map

is an isomorphism. Applying this to shows that the Yoneda embedding

is fully faithful.

Given a functor , write for the comma category whose objects are pairs where and is a morphism (natural transformation) in . A morphism is a morphism such that the triangle

(1)

of natural transformations commutes, where denotes postcomposition by . Note that is again a small category, and there is a forgetful functor taking to .

By the Yoneda lemma, the category is isomorphic (not just equivalent!) to the category of pairs with and with morphisms given by morphisms in such that . It’s convenient to keep both points of view.

Lemma. Let be a small category, and let be a functor.

1. The object is naturally a cocone under via the morphisms .
2. This cocone makes the colimit of the diagram of representable functors.

In particular, any presheaf on a small category is a colimit of representable presheaves.

Proof. A cocone under is a presheaf with a natural transformation to the constant diagram with value . This means every is taken to a natural transformation , such that for any morphism , the square

commutes. By the Yoneda lemma, such a datum corresponds to an association of elements for all such that for every with , we have .

(1) To make a cocone under , simply take .

(2) Given any other cocone under , define the natural transformation by

Naturality follows since implies . It is clear that is the unique natural transformation of cocones under , showing that is the colimit.

One can also easily rewrite this argument in terms of natural transformations . For instance, the universal cocone is the natural transformation of functors given on by . Naturality of follows at once from (1). But checking that this thing is universal is a bit more tedious in this language.

Example. A standard example where this point of view is useful is simplicial sets. Let be the category of finite nonempty totally ordered sets, with (weakly) monotone increasing functions as morphisms. A simplicial set is a functor , and we often think of them as combinatorial models for topological spaces. The representable ones are the standard -simplices , where is the totally ordered set for .

If is a simplicial set, its value at is called the -simplices of . By the Yoneda lemma, this is . Then the story above is saying that a simplicial set is the colimit over all its -simplices for . This is extremely useful, as many arguments proceed by attaching simplices one at a time.

# A strange contractible space

Here’s a strange phenomenon that I ran into when writing a MathOverflow answer a few years ago.

Lemma. Let be a set endowed with the cofinite topology, and assume is path connected. Then is contractible.

The assumption is for example satisfied when , for then any injection is a path from to . Path connectedness of cofinite spaces is related to partitioning the interval into disjoint closed subsets; see the remark below for some bounds on the cardinalities.

Proof. The result is trivial if is finite, for then both are equivalent to . Thus we may assume that is infinite. Choose a path from some to some . This induces a continuous map . Choose a bijection

and extend to a map by and for all . Then is continuous: the preimage of is if , and if , both of which are closed. Thus is a homotopy from to the constant map , hence a contraction.

I would love to see an animation of this contraction as goes from to … I find especially the slightly more direct argument for given here elusive yet somehow strangely visual.

Remark. If is countable (still with the cofinite topology), then is path connected if and only if . In the finite case this is clear (because then is discrete), and in the infinite case this is a result of Sierpiński. See for example this MO answer of Timothy Gowers for an easy argument.

There’s also some study of path connectedness of cofinite topological spaces of cardinality strictly between and , if such cardinalities exist. See this MO question for some results. In particular, it is consistent with ZFC that the smallest cardinality for which is path connected is strictly smaller than .

# Internal Hom

This is an introductory post about some easy examples of internal Hom.

Definition. Let be a symmetric monoidal category, i.e. a category with a functor that is associative, unital, and commutative up to natural isomorphism. Then an internal Hom in is a functor

such that is a left adjoint to for any , i.e. there are functorial isomorphisms

Remark. In the easiest examples, we typically think of as ‘upgrading to an object of ‘:

Example. Let be a commutative ring, and let be the category of -modules, with the tensor product. Then with its natural -module structure is an internal Hom, by the usual tensor-Hom adjunction:

The same is true when is the category of -bimodules for a not necessarily commutative ring .

However, we cannot do this for left -modules over a noncommutative ring, because there is no natural -module structure on for left -modules and . In general, the tensor product takes an -bimodule and a -bimodule and produces an -bimodule . Taking gives a way to tensor a right -module with a left -module, but there is no standard way to tensor two left -modules, let alone equip it with the structure of a left -module.

Example. Let . Then is naturally a set, making it into an internal Hom for :

When is the categorical product , the internal (if it exists) is usually called an exponential object, in analogy with the case above.

Example. Another example of exponential objects is from topology. Let be the category of locally compact Hausdorff topological spaces. Then the compact-open topology makes into an internal Hom of topological spaces. (There are mild generalisations of this beyond the compact Hausdorff case, but for an arbitrary topological space the functor does not preserve colimits and hence cannot admit a right adjoint.)

Example. An example of a slightly different nature is chain complexes: let be a commutative ring, and let be the category of cochain complexes

of -modules (meaning each is an -module, and the are -linear maps satisfying ). Homomorphisms are commutative diagrams

and the tensor product is given by the direct sum totalisation of the double complex of componentwise tensor products.

There isn’t a natural way to ‘endow with the structure of a chain complex’, but there is an internal Hom given by

with differentials given by

Then we get for example

since a morphism is given by an element such that , i.e. , meaning that is a morphism of cochain complexes.

Example. The final example for today is presheaves and sheaves. If is a topological space, then the category of abelian sheaves on has an internal Hom given by

with the obvious transition maps for inclusions of open sets. This is usually called the sheaf Hom. A similar statement holds for presheaves.

# Not every open immersion is an open immersion

An immersion (or locally closed immersion) of schemes is a morphism that can be factored as , where is a closed immersion and is an open immersion. If it is moreover an open morphism, it need not be an open immersion:

Example. Let be a nonreduced scheme, and let be the reduction. This is a closed immersion, whose underlying set is the entire space. Thus, it is a homeomorphism, hence an open morphism. It is not an open immersion, for that would force it to be an isomorphism.

Remark. However, every closed immersion is a closed immersion; see Tag 01IQ.

# Higher pushforwards along finite morphisms

This post is about one of my favourite answers I have given on MathOverflow, although it seems to have gone by mostly unnoticed. In the post, Qixiao asks (essentially) the following:

Question. If is a finite morphism of schemes, is the pushforward exact?

Note that this is true on the subcategory of quasicoherent sheaves because affine morphisms have no quasicoherent higher pushforwards. Also, in the étale topology the pushforward along a finite morphism is exact on the category of all abelian sheaves; see e.g. Tag 03QP.

However, we show that the answer to the question above is negative.

Example. Let be the spectrum of a DVR , let be a finite extension of domains such that has exactly two primes above , and let . For example, and , or and if you prefer a more geometric example.

By my previous post, the global sections functor is exact. If the same were true for , then the global sections functor on would be exact as well. Thus, it suffices to prove that this is not the case, i.e. to produce a surjection of sheaves on such that the map on global sections is not surjective.

The topological space of consists of closed points and a generic point . Let and ; then is open and is closed. Hence, for any sheaf on , we have a short exact sequence (see e.g. Tag 02UT)

where and are the inclusions. Let be the constant sheaf ; then the same goes for and . Then the map

is given by the diagonal map , since is connected by has two connected components. This is visibly not surjective.

# Cohomology of a local scheme

This is a continuation of my previous post on local schemes. Here is a ridiculous lemma.

Lemma. Let be a local scheme, and let be any abelian sheaf on . Then for all .

Proof. It suffices to show that the global sections functor is exact. Let be a surjection of abelian sheaves on , and let be a global section. Then can be lifted to a section of in an open neighbourhood of . But the only open neighbourhood of is . Thus, can be lifted to a section of .

What’s going on is that the functors and are naturally isomorphic, due to the absence of open neighbourhoods of .

Remark. It seems believable that there are suitable site-theoretic versions of this lemma as well. For example, a strictly Henselian local ring has no higher cohomology in the étale topology. The argument is essentially the same: every open neighbourhood of the closed point has a section; see e.g. the proof of Tag 03QO.

# Local schemes

Consider the following definition. It seems to be standard, although I have not found a place where it is actually spelled out in this way.

Definition. A pointed scheme is local if is contained in every nonempty closed subset of .

Example. If is a local ring, then is a local scheme. Indeed, is contained in every nonempty closed subset , because every strict ideal is contained in .

We prove that this is actually the only example.

Lemma. Let be a local scheme. Then is affine, and is a local ring whose maximal ideal corresponds to the point .

Proof. Let be an affine open neighbourhood of . Then the complement is a closed set not containing , hence . Thus, is affine. Let . Let be a maximal ideal of ; then . Since this contains , we must have , i.e. corresponds to the (necessarily unique) maximal ideal .