nLab
compactum

Contents

Idea

A compactum, or compact Hausdorff space, is a space in which every limit that should exist does exist and does so uniquely.

One can define this literally for topological spaces, or in terms of convergence, or for locales; although these are all different contexts, the resulting notion of compactum is (at least assuming the axiom of choice) always the same. Interestingly, there is even an algebraic definition, not one that uses only finitary operations, but one which uses a monad.

Definitions

If you know what a compact space is and what a Hausdorff space is, then you know what a compact Hausdorff space is, so let's be fancy. (Full justifications will be provided in section on compacta as algebras.)

Given a set SS, let βS\beta S be the set of ultrafilters on SS. Note that β\beta is an endofunctor on Set; every function f:STf: S \to T induces a function βf:βSβT\beta f: \beta S \to \beta T using the usual application of functions to filters. In fact, β\beta is a monad; it comes with a natural (in SS) unit η S:SβS\eta_S: S \to \beta S, which maps a point xx to the principal ultrafilter that xx generates, and multiplication μ S:ββSβS\mu_S: \beta \beta S \to \beta S, which maps an ultrafilter UU on ultrafilters to the ultrafilter of sets whose principal ultrafilters of ultrafilters belong to UU. That is,

  • AηxxA A \in \eta x \;\Leftrightarrow\; x \in A , so ηx={ASxA} \eta x = \{ A \subseteq S \;|\; x \in A \} ;
  • AμU{FβSAF}U A \in \mu U \;\Leftrightarrow\; \{ F \in \beta S \;|\; A \in F \} \in U .

Then a compactum is simply an algebra for this monad; that is, a set XX together with a function lim:βXX\lim: \beta X \to X, such that

  • each point xx is the limit (lim\lim) of the principal ultrafilter ηx\eta x, and
  • given an ultrafilter UU on ultrafilters, the limit of μU\mu U is the limit of (βlim)U(\beta \lim) U.

It is then a theorem that this lim\lim generates a convergence on SS that is compact, Hausdorff, and topological. The converse, that every compact Hausdorff topological convergence is of this form, is equivalent to the ultrafilter principle.

Every compact Hausdorff space is regular and sober and so defines a compact regular locale. Again, the axiom of choice gives us a converse: every compact regular locale is spatial and so comes from a compactum.

Probably this is also equivalent to the ultrafilter principle, but I need to check.

Note that every compact Hausdorff space (topological or localic) is not only regular but also normal.

Compacta as algebras

Throughout this section, CHCH will be used to denote the category of compact Hausdorff spaces (compacta).

The space of ultrafilters

Let BoolBool be the category of Boolean algebras. The functor hom(,2):Bool opSet\hom(-, \mathbf{2}): Bool^{op} \to Set has a left adjoint P:SetBool opP: Set \to Bool^{op} given by power sets, and we define the ultrafilter monad to be the composite βhom(P,2)\beta \coloneqq \hom(P -, \mathbf{2}).

For a set SS, topologize βS\beta S by declaring a basic open set to be one of the form

A^{FβS:AF}\hat{A} \coloneqq \{F \in \beta S: A \in F\}

for AA a subset of SS. Notice ^\hat{\emptyset} is empty. Indeed, ()^\widehat{(-)} defines a Boolean algebra map

P(S)P(βS)P(S) \to P(\beta S)

so that in particular AB^=A^B^\widehat{A \cap B} = \hat{A} \cap \hat{B}, which immediately implies that the A^\hat{A} form a basis of a topology.

Remark

In fact, ()^\widehat{(-)} is the component at P(S)P(S) of the counit of the adjunction Phom(,2)P \dashv \hom(-, \mathbf{2}), as a morphism in Bool opBool^{op}. If f:STf: S \to T is a function, the commutativity of the naturality square

P(T) ()^ P(βT) P(f) P(βf) P(S) ()^ P(βS)\array{ P(T) & \stackrel{\widehat{(-)}}{\to} & P(\beta T) \\ \mathllap{P(f)} \downarrow & & \downarrow \mathrlap{P(\beta f)} \\ P(S) & \overset{\widehat{(-)}}{\to} & P(\beta S) }

implies that if U=B^P(βT)U = \hat{B} \in P(\beta T) is a basic (cl)open, then so is (βf) 1(U)=P(βf)(B^)=f 1(B)^(\beta f)^{-1}(U) = P(\beta f)(\hat{B}) = \widehat{f^{-1}(B)}. It then follows that βf\beta f is continuous.

These results show that the monad β:SetSet\beta: Set \to Set lifts through the forgetful functor U:TopSetU: Top \to Set.

The unit of the monad β\beta is given componentwise by functions

prin X:XβXprin_X: X \to \beta X

where prin Xprin_X takes xXx \in X to the principal ultrafilter

prin X(x)={AP(X):xA}.prin_X(x) = \{A \in P(X): x \in A\}.

It is evident that prin Xprin_X is injective.

Proposition

The injection prin X:XβXprin_X: X \to \beta X exhibits XX as a dense subset of βX\beta X.

Proof

If A^\hat{A} is a basic open neighborhood containing an ultrafilter FF, then AA is nonempty and hence contains some xXx \in X, which is to say Aprin X(x)A \in prin_X(x) or that prin X(x)A^prin_X(x) \in \hat{A}.

Ultrafilters form a compactum

Proposition

βS\beta S is Hausdorff.

Proof

Let F,GF, G be distinct ultrafilters, so there is ASA \subseteq S with AFA \in F and ¬AG\neg A \in G. Then A^\hat{A} and ¬A^\widehat{\neg A} are disjoint neighborhoods which contain FF and GG respectively.

Proposition

βS\beta S is compact.

Proof

It is enough to show that if 𝒪\mathcal{O} is a collection of opens such that the union of any finite subcollection is a proper subset, then the union of 𝒪\mathcal{O} is also proper.

If 𝒪\mathcal{O} covers UU, it admits a refinement by basic clopens also covering UU, and thus we may assume WLOG that 𝒪\mathcal{O} consists of basic clopens A^\hat{A}. If every finite union of elements of 𝒪\mathcal{O} is a proper subset of βS\beta S, then every finite intersection ¬A 1^¬A n^\widehat{\neg A_1} \cap \ldots \cap \widehat{\neg A_n} is nonempty, so that the ¬A\neg A generate a filter, which is contained in some ultrafilter FF. This FF lies outside the union of all the A^\hat{A}’s.

Convergence

Definition

Let XX be a topological space, and FF an ultrafilter on the underlying set UXU X. We say FF converges to a point xx (in symbols, FxF \rightsquigarrow x) if the neighborhood filter N xN_x of xx is contained in FF.

Convergence defines a relation ξ\xi from β(UX)\beta(U X) to UXU X.

Proposition

If XX is Hausdorff, then the relation ξ\xi is well-defined, or functional (i.e., there is at most one point to which a given ultrafilter FF converges).

Proof

If xyx \neq y, then there are disjoint neighborhoods UU, VV of xx and yy. We cannot have both UFU \in F and VFV \in F (otherwise =UV\emptyset = U \cap V would be an element of FF), so at most one of the neighborhood filters N x,N yN_x, N_y can be contained in FF.

Proposition

If XX is compact, then the relation ξ\xi is total (i.e., there exists a point to which a given ultrafilter FF converges).

Proof

If not, then for each xXx \in X there is an open neighborhood U xU_x that does not belong to FF. Then ¬U xF\neg U_x \in F. Some finite number of neighborhoods U x 1,,U x nU_{x_1}, \ldots, U_{x_n} covers XX. Then ¬U x 1¬U x n=F\neg U_{x_1} \cap \ldots \cap \neg U_{x_n} = \emptyset \in F, which is a contradiction.

Proposition

If XX is compact Hausdorff, then the function ξ:β(UX)X\xi: \beta(U X) \to X is continuous.

Proof

Let UU be an open neighborhood of xXx \in X; we must show that ξ 1(U)\xi^{-1}(U) contains an open neighborhood of any of its points (i.e., ultrafilters FF such that FxF \rightsquigarrow x). Since XX is T 3T_3 (Hausdorff regular), we may choose a neighborhood VN xV \in N_x whose closure V¯\bar{V} is contained in UU. Then V^\hat{V} is an open neighborhood of FF in β(UX)\beta(U X), and we claim V^ξ 1(U)\hat{V} \subseteq \xi^{-1}(U).

For this, we must check that if GV^G \in \hat{V} and GyG \rightsquigarrow y, then yUy \in U. But if y¬U¬V¯y \in \neg U \subseteq \neg \bar{V}, then ¬V¯N y\neg \bar{V} \in N_y, whence GyG \rightsquigarrow y implies ¬V¯G\neg \bar{V} \in G. This contradicts GV^G \in \hat{V}, i.e., contradicts VGV \in G, since V¬V¯=V \cap \neg \bar{V} = \emptyset.

Spaces of ultrafilters are universal

Proposition

If SS is a set and XX is a compact Hausdorff space, then any function f:SXf: S \to X can be extended (along prin S:SβSprin_S: S \to \beta S) to a continuous function f^:βSX\hat{f}: \beta S \to X.

Proof

We define f^\hat{f} to be the composite

βSβ(f)β(UX)ξX\beta S \stackrel{\beta(f)}{\to} \beta (U X) \stackrel{\xi}{\to} X

where β(f)\beta(f) is continuous by Remark 1 and ξ\xi is continuous by Proposition 6. It remains to check that the following diagram is commutative:

S f UX prin S prin UX 1 UX βS β(f) β(UX) ξ UX.\array{ S & \stackrel{f}{\to} & U X & & \\ \mathllap{prin_S} \downarrow & & \mathllap{prin_{U X}} \downarrow & \searrow \mathrlap{1_{U X}} & \\ \beta S & \underset{\beta (f)}{\to} & \beta (U X) & \underset{\xi}{\to} & U X. }

The square commutes by naturality of prinprin, and commutativity of the triangle simply says that the ultrafilter prin UX(x)prin_{U X}(x) converges to xx, or that N xprin(x)N_x \subseteq prin(x), which reduces to the tautology that xVx \in V for every neighborhood VN xV \in N_x.

Theorem

For any set SS, the function prin S:SβSprin_S: S \to \beta S is universal among functions from SS to compact Hausdorff spaces. Hence the functor F:SetCHF: Set \to CH that takes SS to the compact Hausdorff space βS\beta S is left adjoint to the forgetful functor CHSetCH \to Set.

Proof

Proposition 7 shows that for any function f:SUXf: S \to U X to a compact Hausdorff space, there exists continuous g:βSXg: \beta S \to X such that gprin S=fg \circ prin_S = f. All that remains is to establish uniqueness of such gg. But if two maps g,g:βSXg, g': \beta S \to X to a Hausdorff space XX agree on a dense subspace, in this case the subspace prin S:SβSprin_S : S \hookrightarrow \beta S by Proposition 1, then they must be equal. Indeed, the pullback of the closed diagonal defines a closed subspace DD of βS\beta S,

D X δ X βS g,g X×X,\array{ D & \to & X \\ \downarrow & & \downarrow \mathrlap{\delta_X} \\ \beta S & \underset{\langle g, g' \rangle}{\to} & X \times X, }

and DD contains a dense subspace SS, therefore D=βSD = \beta S; i.e., the equalizer of gg and gg' is all of βS\beta S, hence these two maps are equal.

Compact Hausdorff spaces are monadic over sets

We recall hypotheses of Beck’s precise monadicity theorem: a functor U:CDU: C \to D is monadic if and only if

  1. UU has a left adjoint,

  2. UU reflects isomorphisms: a morphism f:XYf: X \to Y of CC is an isomorphism if Uf:UXUYU f: U X \to U Y is an isomorphism in DD,

  3. DD has, and UU preserves, coequalizers of parallel pairs that are UU-split. (We say

    XgfYX \stackrel{\overset{f}{\to}}{\underset{g}{\to}} Y

    is UU-split if there is a coequalizer

    UXUgUfUYhZU X \stackrel{\overset{U f}{\to}}{\underset{U g}{\to}} U Y \stackrel{h}{\to} Z

    that is split in DD: there exists i:ZUYi: Z \to U Y and j:UYUXj: U Y \to U X such that Uhi=1 ZU h \circ i = 1_Z, Ugj=ihU g \circ j = i \circ h, and Ufj=1 UYU f \circ j = 1_{U Y}.)

In the case where D=SetD = Set, we have the following useful lemma:

Lemma

Suppose given a coequalizer in SetSet

XgfYhZX \stackrel{\overset{f}{\to}}{\underset{g}{\to}} Y \stackrel{h}{\to} Z

split by i:ZYi: Z \to Y, j:YXj: Y \to X (so that fj=1 Yf j = 1_Y, hi=1 Zh i = 1_Z, gj=ihg j = i h). Let RY×YR \hookrightarrow Y \times Y be the image of f,g:XY×Y\langle f, g \rangle : X \to Y \times Y, and let p 1,p 2:EY×Y\langle p_1, p_2 \rangle : E \to Y \times Y be the equivalence relation given by the kernel pair (p 1,p 2)(p_1, p_2) of hh. Then E=RR opE = R \cdot R^{op}, the relational composite given by taking the image of the span composite R× YR opY×YR \times_Y R^{op} \to Y \times Y.

Proof

Clearly RER \subseteq E since hf=hgh f = h g, and we have R opE op=ER^{op} \subseteq E^{op} = E and RR opEEER \cdot R^{op} \subseteq E \cdot E \subseteq E by symmetry and transitivity of EE. In the other direction, suppose (y 1,y 3)E(y_1, y_3) \in E, so that h(y 1)=h(y 3)h(y_1) = h(y_3). Put x=j(y 1)x = j(y_1) and x=j(y 3)x' = j(y_3) (so that f(x)=y 1f(x) = y_1 and f(x)=y 3f(x') = y_3), and put y 2=g(x)y_2 = g(x). Clearly then (y 1,y 2)R(y_1, y_2) \in R. Moreover,

y 2=g(x)=gj(y 1)=ih(y 1)=ih(y 3)=gj(y 3)=g(x)y_2 = g(x) = g j(y_1) = i h(y_1) = i h(y_3) = g j(y_3) = g(x')

so that (y 3,y 2)R(y_3, y_2) \in R, or (y 2,y 3)R op(y_2, y_3) \in R^{op}. Hence (y 1,y 3)RR op(y_1, y_3) \in R \cdot R^{op}, and we have shown ERR opE \subseteq R \cdot R^{op}.

Theorem

The forgetful functor U:CHSetU: CH \to Set is monadic.

Proof

By theorem 1, UU has a left adjoint. Since bijective continuous maps between compact Hausdorff spaces are homeomorphisms, we have that UU reflects isomorphisms. Finally, suppose (f,g)(f, g) is a UU-split pair of morphisms XYX \to Y in CHCH; let h:YZh: Y \to Z be their coequalizer in TopTop, given by a suitable quotient space. Being a quotient of a compact space, ZZ is compact. Since CHCH is a full subcategory of TopTop, the map hh is a coequalizer in CHCH once we prove the following claim:

  • Claim: ZZ is Hausdorff.

Furthermore, since the forgetful functor TopSetTop \to Set has a right adjoint (given by taking indiscrete topologies on sets), the underlying function of hh (again denoted hh) is the coequalizer of (Uf,Ug)(U f, U g) in SetSet, so that UU would preserve the claimed coequalizer.

In other words, to complete the proof, it suffices to verify the claim. Letting p 0,p 1:EYp_0, p_1: E \stackrel{\to}{\to} Y be the kernel pair of hh in TopTop, to show Z=Y/EZ = Y/E is Hausdorff, it suffices to prove that the equivalence relation p 0,p 1:EY×Y\langle p_0, p_1 \rangle : E \to Y \times Y in TopTop is closed. Let RY×YR \hookrightarrow Y \times Y be the image of f,g:XY×Y\langle f, g \rangle: X \to Y \times Y. By lemma 1, the subset EE of UY×UYU Y \times U Y coincides with the subset RR opUY×UYR \cdot R^{op} \subseteq U Y \times U Y. Now RR is the image of the compact space XX under the continuous map f,g\langle f, g \rangle, so RR is a closed subset of Y×YY \times Y. Similarly R opR^{op} is a closed subset of Y×YY \times Y. Under their subspace topologies, their fiber product R× YR opR \times_Y R^{op} is compact, and so its image RR opR \cdot R^{op} under the (continuous) span composite R× YR opY×YR \times_Y R^{op} \to Y \times Y is also closed. This completes the proof.

In weak foundations

In the absence of the axiom of choice, and especially in constructive mathematics, the best definition of compactum seems to be a compact regular locale. That is, it is the category of compact regular locales that has all of the nice properties, forming a nice category of spaces, and that has the desired examples, such as the unit interval. (See the discussion at Tychonoff theorem for an example of how the category of compact Hausdorff topological spaces might fail to be nice; see Frank Waaldijk’s PhD thesis (pdf) for a thorough discussion of what is needed to make the unit interval a compact Hausdorff topological space.)

The monadic definition, in particular, falls quite flat without some form of the axiom of choice; even excluded middle and COSHEP are powerless here. In fact, it is quite consistent to assume that every ultrafilter is principal (a strong denial of the ultrafilter principle), in which case β\beta is the identity monad. Then a compactum would be just a set if that were the definition used.

On the other hand, it is the monadic definition that gives an algebraic category with a nice relationship to Set. Without the ultrafilter principle, there is no reason to think that the set-of-points functor from compact regular locales to sets is even continuous.

Stone–Čech compactification

By general nonsense, every βS\beta S, regarded as a free β\beta-algebra, is a compactum, and the functor

β:SetComp\beta: Set \to Comp

is left adjoint to the forgetful functor CompSetComp \to Set. Assuming the ultrafilter principle, this functor extends to a functor β:TopComp\beta: Top \to Comp (identifying a set with its discrete space) that is left adjoint to the forgetful functor CompTopComp \to Top. This is the Stone–Čech compactification functor (N.B.: for many authors, Stone–Čech compactification refers to the restriction of this functor to Tychonoff spaces XX, which are precisely those spaces where the unit XβXX \to \beta X is an embedding so that we have a compactification in the technical sense).

A classical construction of the Stone–Čech compactification starts with the unit interval I=[0,1]I =[0, 1] and proceeds to the codensity monad induced from the functor

hom(,I):Top opSet.\hom(-, I) \colon Top^{op} \to Set.

The monad is given on objects by XI hom(X,I)X \mapsto I^{\hom(X, I)}; this lands in compact Hausdorff spaces. Let X¯\bar{X} be the closure of the image of the unit u X:XI hom(X,I)u_X: X \to I^{\hom(X, I)}; under the ultrafilter principle, X¯\bar{X} is compact Hausdorff.

Proposition

If XX is a Tychonoff space, then the unit u X:XI hom(X,I)u_X: X \to I^{\hom(X, I)} is a subspace embedding, so that II is a cogenerator in the category of Tychonoff spaces. (In particular, u Cu_C is an embedding if CC is compact Hausdorff, so II is also a cogenerator in the category of compact Hausdorff spaces.)

The proof is essentially Urysohn's lemma; see also related discussion at Tychonoff space and at uniform space (noting that compact Hausdorff spaces are uniform spaces for a unique uniformity).

Theorem

The natural map i X:XX¯i_X: X \to \bar{X} is universal among maps from XX to compact Hausdorff spaces, thus giving a left adjoint TopCompTop \to Comp to the (fully faithful) forgetful functor U:CompTopU: Comp \to Top.

Proof

Let f:XCf: X \to C be a map, where CC is a compact Hausdorff space. Since II is a cogenerator in the category of compact Hausdorff spaces, the unit for the codensity monad M IM_I,

u C:CI hom(C,I),u_C: C \to I^{\hom(C, I)},

is a continuous injection (and hence a closed subspace embedding, since CC is compact Hausdorff). Let f^=M I(f)\hat{f} = M_I(f), and consider the pullback square

f^ 1(C) I hom(X,I) π f^ C u C I hom(C,I).\array{ \hat{f}^{-1}(C) & \to & I^{\hom(X, I)} \\ \mathllap{\pi} \downarrow & & \downarrow \mathrlap{\hat{f}} \\ C & \underset{u_C}{\to} & I^{\hom(C, I)}. }

From an evident naturality square for the unit uu, we have a map h:Xf^ 1(C)h: X \to \hat{f}^{-1}(C), i.e., the map u X:XI hom(X,I)u_X: X \to I^{\hom(X, I)} factors through the closed subspace f^ 1(C)I hom(X,I)\hat{f}^{-1}(C) \hookrightarrow I^{\hom(X, I)}. Therefore hh factors as

Xi XX¯f^ 1(C)X \stackrel{i_X}{\to} \bar{X} \subseteq \hat{f}^{-1}(C)

and since πh=f\pi \circ h = f, we conclude that ff factors through i Xi_X. And moreover, there is at most one k:X¯Ck: \bar{X} \to C such that ki X=fk \circ i_X = f, because i Xi_X maps XX onto a dense subspace of X¯\bar{X}, and dense subspaces are epic in the category of Hausdorff spaces. This completes the proof.

We have a similar Stone–Čech compactification functor LocCompLoc \to Comp; we do not need the ultrafilter principle here if CompComp is defined in terms of locales.

Category of compacta

The category CompComp of compact Hausdorff spaces and continuous maps is

From the first two properties, it follows that CompComp is a pretopos, meaning that CompComp enjoys the same finitary exactness properties that hold in a topos; in particular, first-order intuitionistic logic may be enacted within CompComp.

Revised on December 3, 2013 13:23:24 by Toby Bartels (64.89.53.109)