Lemma 34. Letκbe strongly compact andXa space.
1. If U ⊆ X is open, Hausdorff and locally κ-compact, then ιX[U] is open, c U is a homeomorphism andιX[{U]is disjoint fromιX[U].
2. Ifa,b ⊆ Xare closed and disjoint and Xis normal, thenιX[a]andιX[b]have disjoint closures.
Proof. (1): Letq :X → ωUbe the identity onUand map everything else to the added point p, as in Lemma 29. Thenq Uis continuous and injective, and sinceβq◦ιX = q, the map ιX Uis injective, too. Also, its image is disjoint fromιX[{U], because the latter is mapped to {p}. It remains to prove thatιX[V] = (βq)−1[V]for every openV ⊆ U, which will imply that it is in fact open.
So assumex ∈ (βq)−1[V]. Let W ⊆ V be aκ-compact neighborhood of βq(x). Since ιX[X]
is dense, xis in the closure of ιX[X]∩(βq)−1[W]. But that is just equal to ιX[W], which is κ-compact and therefore already closed. Hencexis in the image ofW ⊆V.
(2): Letb0 = b. IfXis normal, there is for everyn∈ ωabn+1such thatbn ⊆int(bn+1)and bn+1∩a = ∅. Thenc = S
n∈ωbn = S
n∈ωint(bn)is a clopen set withb⊆candc∩a= ∅.
Thusf(x) = 1 iffx ∈cdefines a continuousf :X→ Dwithf[a]⊆ {0}andf[b]⊆1. SinceD isκ-compact, there is a mapβf : βX → Dwithβf◦ιX = f. In particular, ιX[a]is a subset of the closed set(βf)−1[0]andιX[b]⊆(βf)−1[1], so they have disjoint closures.
2.3. CATEGORIES OFκ-TOPOLOGICAL SPACES 49
g◦f= h◦fimpliesg = h. f :X → Y is called an isomorphism, andXandYareisomorphic, if there is ag : Y → X, theinverseoff, withg◦f = idXandf◦g = idY. Aninitial objectis an objectXsuch that for everyY there is a unique morphismf :X→ Y, and aterminal object is one such that for eachY there is exactly one morphism f : Y → X. Since all morphisms between initial objects are necessarily isomorphisms, all initial objects are isomorphic, and the same goes for terminal objects.
Given a fixed objectZ, morphisms with domain Z can themselves be considered objects of a category, where the morphisms from g : Z → X to h : Z → Y are all f : X → Y for whichh = f◦g. Similarly morphisms with codomainZform a category, in whichf : X → Y is a morphism from g : X → Z to h : Y → Z iff h◦f = g. In these two senses, we will also speak ofinitialandterminal morphisms. For example, theκ-ˇCech-Stone compactification ιX :X→ βXis initial among the maps fromXtoκ-compact Hausdorff spaces.
A functorF from a category Cto a category Dis a mapping that assigns to each object re-spectively arrow of C an object respectively arrow of D such that whenever f : X → Y, F(f) : F(X) → F(Y), and such thatF(idX) = idF(X) andF(f◦g) = F(f)◦F(g)for allX,fand g. A natural transformation φ from a functorF to a functor G, both from C to D, maps to each objectX of C a morphism φ(X) : F(X) → G(X) such that for every f : X → Y in C, φ(Y)◦F(f) = G(f)◦φ(X). It is customary to omit the brackets after functors and writeFffor F(f)andFXforF(X).
In the categories we will encounter, the objects are sets with added structure, the morphisms are maps which preserve that structure and ◦ is the composition of functions. In particular, Hom(X,Y)will always be a set. In this section, we will mainly be concerned with the category Top whose objects areκ-topological spaces and whose morphisms are continuous maps, as well as with its subcategory of only theκ-compact Hausdorff spaces.
Let I be adirected set, that is, a partially ordered set in which two elements always have a common upper bound. Adirected system in a categoryCis a familyhXi,fi,jii6j∈I of objects Xi and morphismsfi,j : Xi → Xj for alli 6 j, indexed byI, such that wheneveri 6 j 6 k, fi,k = fj,k◦fi,j, and such thatfi,i = idXi for alli∈ I. Aconefrom the familyhXi,fi,ji(toY) is an objectYand a family of morphismsgi :Xi → Ysuch that for alli6 j,gj◦fi,j = gi. It is adirect limitofhXi,fi,jiif it has the following universal property: For everyZand every cone hZ,hiii∈IfromhXi,fi,jitoZ, there is a unique morphismh :Y → Zsuch thath◦gi = hifor alli ∈I. All direct limits of a given family are isomorphic, so we also speak ofthedirect limit and write:
hY,gii = lim
→
ChXi,fi,ji
In the category ofκ-topological spaces and continuous maps, a direct limit can be constructed as follows: Forx∈ Xiandy ∈Xi0, letx ∼yiff there is aj > i,i0such thatfi,j(x) = fi0,j(y).
LetY = S
i∈IXi/ ∼be the quotient and letgi : Xi → Y be the quotients of the embeddings.
ThenhY,giiin fact has the universal property of a direct limit, because whenever hZ,hiii∈I is a cone from hXi,fi,ji to Z, the union S
hi : S
i∈IXi → Z factors through ∼, defining a corresponding maph:Y → Z.
Aninverse systeminC is a familyhXi,fi,jii6j∈I of objects Xiand morphisms fi,j : Xj → Xi (note the reversed direction) for alli 6 j, indexed by a directed setI, such that whenever i 6j 6 k,fi,k = fi,j◦fj,k, andfi,i = idXi for alli ∈ I. Aconeto that family (from Y) is an objectY and a family of morphismsgi : Y → Xisuch that for alli 6j,gi = fi,j◦gj. It is an
inverse limitofhXi,fi,jiif it has the following universal property: For every conehZ,hiii∈I to hXi,fi,ji, there is a unique morphismh : Z → Y such that gi◦h = hifor all i ∈ I. Again, all inverse limits of a given family are isomorphic, so we also speak ofthe inverse limit and write:
hY,gii = lim
←
ChXi,fi,ji
For κ-topological spaces, an inverse limit can also be constructed: Let Y be the subspace of Q
i∈IXi given by those elements (xi)i∈I such that fi,j(xj) = xi for all j > i, and let gi :Y → Xibe the projection onto thei-th component. If nowhZ,hiii∈Iis a cone tohXi,fi,ji, thenh :Z→ Y, h(z) = (hi(z))i∈I witnesses the universal property.
Note that if theXiare Hausdorff spaces, thenY is a closed subspace of the product and also Hausdorff. If in addition the product isκ-compact,Y is, too. In that case,Yis also the inverse limit in the category ofκ-compact Hausdorff spaces.
Unfortunately the matter of separation properties is more complicated in the context of direct limits, where even κ-compact Hausdorff spaces can have non-Hausdorff limits5. But still, direct limits may exist in the category ofκ-compact Hausdorff spaces: If every space has a κ-ˇCech-Stone compactification – which is the case for κ = ω or strongly compact κ – then theκ-ˇCech-Stone compactification of the direct limit in the category of κ-topological spaces is the direct limit in the category ofκ-compact Hausdorff spaces. This is because the maps fromβXtoY correspond bijectively to the maps fromXto Y for everyκ-compact Hausdorff spaceYand everyκ-topological spaceX, which makes theκ-ˇCech-Stone compactification the left adjoint of the inclusion functor:
Two functors F : C → D and G : D → C between categories C and Dare called adjoint, whereFis theleft adjointofGandGis theright adjointofF, if for all objectsXofCandY of D, there is a bijection
ΦX,Y :Hom(FX,Y) → Hom(X,GY) such that for everyf :X0 → X,g :Y → Y0andh:FX → Y,
ΦX0,Y0(g◦h◦Ff) = Gg◦ΦX,Y(h)◦f.
If that is the case,Fpreserves direct limits andGpreserves inverse limits.
We call a subset Kof a partial order P ν-closedif all ν-small subsets A ⊆ I which have an upper bound inP also have an upper bound inK. A partial order in which ν-few elements always have an upper bound is calledν-directed.
5For example let X0 = {0,11,12,13,...} ⊆ R and let Xn be the quotient where the n greatest fractions are identified. Then eachXnis Hausdorff, but the direct limit is the Sierpi´nski space.
2.3. CATEGORIES OFκ-TOPOLOGICAL SPACES 51
Lemma 35. LetCbe a category in which inverse limits exist. Letf:X→ Xebe an isomorphism inC, where
hX,giii∈I = lim
←hXi,gijii,j∈I and hX,egeiii∈eI = lim
←hXei,geijii,j∈eI
are inverse limits of inverse systems, where allgiandegiare epimorphisms. Assume that for everyi ∈I, there is anei∈eIand a morphisms :Xe
ei → Xiwiths◦ge
ei= gi◦f−1, and for everyei ∈eI, there is ani∈ Iand a morphismes :Xi → Xe
eiwithes◦gi= ge
ei◦f.
Xioo gi X
f
Xi
es
gi X
oo
f Xe
ei s
OO
Xe
ge
ei
oo eX
eioo geei Xe
LetJ ⊆ I andeJ ⊆eI have cardinality 6 νfor some inaccessible cardinal ν. Then there are ν-closed subsetsK ⊇JofIandKe ⊇eJofeIof cardinality6ν, such that the inverse limits
hY,eiii∈K = lim
←hXi,gijii,j∈K and heY,eeiii∈
Ke = lim
←hXei,egiji
i,j∈Ke
are canonically isomorphic.
Proof. Leteh : I → eI andh :eI → Ibe choices of suitable indices, such that for all i ∈ Iand ei ∈eIthere exist morphisms si : Xe
h(i)e → Xiandes
ei : Xh(ei) → Xe
eiwith si◦eg
h(i)e = gi◦f−1 andes
ei◦gh(ei) = eg
ei◦f.
Letc : Pν(I) → Ibe a choice of upper bounds in I, that is, for eachA ∈ Pν(I), let c(A)be an upper bound of Aif there is one, and an arbitrary element ofI otherwise. Similarly, let ec:Pν(eI)be such a choice for upper bounds ineI. We defineK ⊆IandKe ⊆eIas the closure of JandeJwith respect toh,eh,candec, that is,
K0 = J K00 = eJ
Kα+1 = Kα ∪ h[eKα] ∪ c[Pν(Kα)] eKα+1 = Keα ∪ eh[Kα] ∪ ec[Pν(eKα)]
Kβ = S
α<βKα eKβ = S
α<βKeα
for all ordinalsαand all limit ordinals β. ThenK = Kν = Kν+1 andKe = Keν = Keν+1have cardinality6νand areν-closed and in particular directed.
For alli < jinK,h(i),e eh(j)are elements ofKeand there is an indexme ∈ Kewithme >eh(i),h(j).e
Since in the diagram Xj
gij
Xe
h(j)e sj
oo
X
gj
kk
gi
ss
f //Xe
ge
eh(j)
33
ge
fm //
ge
h(i)e
++
Xe
fm ge
h(j)e mf
OO
ge
h(i)e fm
Xi Xe
h(i)e si
oo
all cells are commutative, all paths fromXtoXiare equivalent, and becausefis an isomor-phism andge
fm : Xe → Xe
mf an epimorphism, it follows thatgij◦sj◦ge
h(j)e mf = si◦ge
h(i)e mf and thus
gij◦sj◦ee
h(j)e = gij◦sj◦ge
h(j)e fm◦eem =si◦eg
h(i)e mf◦ee
mf = si◦ee
h(i)e , so the family of morphismssi◦ee
h(i)e :Ye→ Xiwithi ∈Kis a cone fromYetohXi,gijii,j∈Kand defines via the universal property of the inverse limit a morphisms :Ye→ Y. Symmetrically, there is a corresponding morphismes : Y → Y. It remains to show thate s andes are inverses of each other and again using symmetry, we will only prove that s◦es = idY. Again by the universal property of the inverse limitY, this follows if we can demonstrateei◦s◦es = ei◦idY =eifor alli∈K.
Giveni ∈K, letm∈ Kwithm>i,j, wherej = h(eh(i)). Using the fact thatgm :X → Xm is an epimorphism, we see that the following diagram commutes:
Xi
Xm
gim
22
gjm
++
gm X
oo f //
gi
OO
gj
Xe e
gh(i)e //Xe
h(i)e si
mm
Xj e
sh(i)e
44
Hencesi◦es
h(i)e ◦gjm = gim. Using that andgim◦em = ei, the commutative diagram Y
ei
Ye
oo s
ee
h(i)e
Y
se
oo
ej
em
%%Xi eX
h(i)e si
oo Xje
s
h(i)e
oo Xmgjmoo
gim
jj
shows thatei◦s◦es = ei.
2.3. CATEGORIES OFκ-TOPOLOGICAL SPACES 53
Lemma 36. LethXi,gijii,j∈IandhXej,geijii,j∈eIbeκ-directed systems ofκ-small discrete spaces, allgiandgej surjective, and let their inverse limits
hX,giii∈I = lim
←hXi,gijii,j∈I and hX,egeiii∈I = lim
←hXei,geiji be homeomorphic. Then all subsystems ofhXi,gijii,j∈I andhXei,geiji
i,j∈eI with at most κ ob-jects can be extended toκ-closed subsystems of size6κwhose inverse limits are homeomor-phic.
Proof. Letf : X → Xebe a homeomorphism. It suffices to prove the assumptions of Lemma 35. By symmetry, we only have to findeiandsi :Xe
ei → Xifor any giveni.
First we note that the preimages of the formge−1j [a]are not only aκ-subbase but actually a base ofX, because ae κ-small union of such sets is of that form itself:
[
α<γ
eg−1j
α [aα] = [
α<γ
ge−1k ◦gejαk−1[aα] = ge−1k
"
[
α<γ
ge−1j
αk[aα]
# ,
wherekis an upper bound of thejα. In particular, every clopen subsetcofXeis an intersection of such sets. But since{c is κ-compact, c is in fact an intersection of κ-few base sets and therefore a base set itself:
c= \
α<γ
ge−1j
α[aα] = \
α<γ
eg−1k ◦gejαk−1[aα] =ge−1k
"
\
α<γ
eg−1j
αk[aα]
#
For everyi ∈I, the setsf◦g−1i [{x}]withx∈ XipartitionXeintoκ-few clopen setsge−1j
x [ax]. If kis an upper bound of{jx | x∈Xi}and we setbx = ge−1j
xk[ax], we havef◦g−1i [{x}] =ge−1k [bx].
Hence we can defineei= kandsi(y) = xfor ally∈bx.
Lemma 37. Let I be a κ-directed set andhgij,Xiii,j∈I an inverse system in the category of κ-topological spaces and assume that everyXi isκ-compact. Let the spaces Xei = Expκ(Xi) and the mapsgeij = Expκ(gij)be its corresponding exponential system. Consider the limits
hX,giii∈I = lim
←hXi,gijii,j∈I and hX,egeiii∈I = lim
←hXei,egiji.
ThenXeis canonically homeomorphic to the space Expκ(X).
Proof. We can wlog assume that allgij (and consequently allgeij) are surjective, because an inverse system of spaces always has the same limit as the corresponding system of subspaces gi[X] = T
j>igij[Xj] ⊆ Xi in which the maps are surjective. Since the Xi areT3, every Xei is Hausdorff and so isX.e
The maps Expκ(gi) : Expκ(X) → Xeicommute with the mapsgfij, so we obtain a continuous map f :Expκ(X) → Xefrom the inverse limit property ofXeand we only have to show that it is bijective.
Firstly, let a,b ∈ Expκ(X) be distinct, wlog let x ∈ a\b. Then for everyy ∈ b, there is an iy ∈ I, such thatgiy(x)6= giy(y). SinceXiis Hausdorff, there are disjoint openUy 3 giy(x) andVy 3 giy(y) separating them. But κ-few sets g−1i
y[Vy] suffice to cover b, and if j is an upper bound of theirκ-few indicesiy, thengj(x)∈/ gj[b], because
giyj◦gj(x) = giy(x) ∈/ Vy = giyj◦gjh g−1i
y[Vy]i
for ally.
As a consequencegj[a]6=gj[b]. Thus the map Expκ(gj) : Expκ(X) →Xej, which equalsgej◦f, mapsaandbto distinct points, implying thatfalso does.
Secondly, letb∈ X. Then for everye i,gei(b)is aκ-compact subset ofXi. We claim that the set
a = \
i∈I
g−1i [gei(b)]
is an element of Expκ(X)andf(a) = b.
Wheneverj > i,
g−1j [gej(b)] ⊆ g−1j [g−1ij [geij◦egj(b)]] = g−1i [gei(b)].
So if aκ-small family of setshg−1i
α [geiα(b)]|α < γiis given, andjis an upper bound of theiα, everyg−1i
α [egiα(b)]is a superset ofg−1j gej(b)
, which is nonempty. Therefore these sets have noκ-small subcocover and becauseXisκ-compact,ais in fact nonempty, soa∈Expκ(X) To provef(a) = b, it suffices to verify thatgej◦f(a) = gej(b)for allj. Butgej◦f = Expκ(gj) and it follows from the definition ofathat Expκ(gj)(a) ⊆ gej(b). To prove the converse, let x /∈Expκ(gj)(a), that is, letg−1j [{x}]be disjoint froma. Sinceg−1j [{x}]isκ-compact there are κ-few setsg−1i [gei(b)]whose intersection withg−1j [{x}]is empty, and becauseI isκ-directed, there exists a single i > j, such that g−1j [{x}]∩g−1i [egi(b)] = ∅. Hence x /∈ gij[gei(b)] = egij(egi(b)) =gej(b).