Elements of monoidal topology
Lecture 6: separation axioms for generalized spaces
Sergejs Solovjovs
Department of Mathematics, Faculty of Engineering, Czech University of Life Sciences Prague (CZU)
Kam´yck´a 129, 16500 Prague - Suchdol, Czech Republic
Abstract
This lecture continues to view (T, V )-categories as generalized spaces and considers the respective generalized
versions of low separation axioms (T0, R0, T1, R1), regulariy, normality, and also extremal disconnectedness.
1. Order separation
Remark 1. Since this lecture considers properties inspired by general topology, given a category (T, V )-Cat,
its objects (resp. morphisms) will be often referred to as (T, V )-spaces (resp. (T, V )-continuous maps).
Definition 2.
(1) Recall from Lecture 2 that given a (T, V )-space (X, a), the V -relation TX
a
// X induces a preorder
⩽ on the set X defined for every x, y ∈ X by x ⩽ y iff k ⩽ a(eX(x), y) (where e is the unit of the monad
T). This preorder is called the underlying preorder induced by a or simply the induced preorder.
(2) A (T, V )-space (X, a) is said to be separated provided that its underlying preorder is a partial order,
i.e., for every x, y ∈ X, if x ⩽ y and y ⩽ x, then x = y.
(3) The full subcategory of the category (T, V )-Cat of separated (T,V)-spaces is denoted (T, V )-Catsep.
Definition 3. A topological space (X, τ), where τ is a topology on the set X, is called a T0-space provided
that for every two distinct points of X, there exists an element of τ containing exactly one of them.
Example 4.
(1) In the category 2-Cat, which is exactly the category Prost of preordered sets and monotone maps,
separated 2-categories are exactly the partially ordered sets (posets, for short).
(2) In the category P+-Cat, which is exactly the category QPMet of quasi-pseudo-metric spaces (generalized
metric spaces in the sense of F. W. Lawvere) and non-expansive maps, separated P+-categories are
quasi-pseudo-metric spaces (X, ρ) such that for every x, y ∈ X, if ρ(x, y)=0 and ρ(y, x)=0, then x=y.
(3) In the category (β, 2)-Cat, which is exactly the category Top of topological spaces and continuous maps,
separated (β, V )-categories are topological spaces (X, τ) such that for every x, y ∈ Y , if the principal
ultrafilter ˙x converges to y, and the principal ultrafilter ˙y converges to x, then x = y. Recall from
Lecture 3 that an ultrafilter x ∈ βX converges to some x ∈ X provided that x contains every U ∈ τ such
that x ∈ U. In view of Definition 3, separated topological spaces are precisely the T0-spaces.
Email address: solovjovs@tf.czu.cz (Sergejs Solovjovs)
URL: http://home.czu.cz/solovjovs (Sergejs Solovjovs)
Preprint submitted to the Czech University of Life Sciences Prague (CZU) January 3, 2022
(4) In the category (β, P+)-Cat, which is exactly the category App of approach spaces and non-expansive
maps, an approach space (X, a) is separated iff for every x, y ∈ X, a( ˙x, y) = 0 and a( ˙y, x) = 0 imply
x = y. Equivalently, in terms of the approach distance δ, δ(x, {y}) = 0 and δ(y, {x}) = 0 imply
x = y, where the approach distance X × PX
δ
−→ [0, ∞] of a P+-category (X, a) is defined by the formula
δ(z, C) = inf{a(y, z) | y ∈ βC} for every z ∈ X and every C ⊆ X.
Proposition 5. For every (T, V )-space (X, a), the following holds.
(1) If (X, a) is Hausdorff, then (X, a) is separated.
(2) If (X, a) is separated, then every (T,V)-continuous map (2 = {0, 1}, ⊤V )
f
−→ (X, a) from a two-element
indiscrete (T, V )-space (recall from Lecture 2 that ⊤V stands for the constant map T2 × 2
⊤V
−−→ V with
value ⊤V ) is constant. If the quantale V is strictly two-sided (k = ⊤V ), then the latter property is
equivalent to (X, a) being separated provided that for every map 2
f
−→ X, it follows that a(Tf(x), f(i)) =
⊤V for every x ∈ T2 such that x ̸∈ {e2(0), e2(1)} and every i ∈ 2.
(3) The full subcategory (T, V )-Catsepof separated (T,V)-spaces is closed under mono-sources in (T, V )-Cat.
Proof.
(1) Recall from Lecture 5 that a (T, V )-space (X, a) is Hausdorff provided that a · a◦
⩽ 1X, which implies,
in particular, that for every x1, x2 ∈ X and every y ∈ TX, if ⊥V < a(y, x1) ⊗ a(y, x2), then x1 = x2.
Given now x, y ∈ X such that x ⩽ y and y ⩽ x, it follows that k ⩽ a(eX(x), y) and k ⩽ a(eX(y), x), i.e.,
⊥V < k = k ⊗ k ⩽ a(eX(x), y) ⊗ a(eX(y), x), which gives y = x by the above Hausdorffness property.
(2) Given a (T, V )-continuous map (2, ⊤V )
f
−→ (X, a), it follows that ⊤V ⩽ f◦
· a · Tf (recall Lecture 5),
which gives k ⩽ ⊤V = ⊤V (e2(0), 1) ⩽ (f◦
· a · Tf)(e2(0), 1) = (recall Lecture 2) = a(Tf(e2(0)), f(1)) =
a((Tf · e2)(0), f(1)) = (since 1Set
e
−→ T is a natural transformation, the diagram
2
f

e2
// T2
T f

X eX
// TX
(1.1)
commutes, i.e., Tf ·e2 = eX ·f) = a((eX ·f)(0), f(1)) = a(eX(f(0)), f(1)), i.e., f(0) ⩽ f(1). In a similar
way, one obtains that f(1) ⩽ f(0), which implies f(0) = f(1), since the (T, V )-space (X, a) is separated.
For the second statement, to show that (X, a) is separated, take x, y ∈ X such that x ⩽ y and y ⩽ x.
Define a map 2
f
−→ X by f(0) = x and f(1) = y. If f is (T, V )-continuous, then (by the assumption)
f is constant, i.e., x = f(0) = f(1) = y. Thus, it is enough to prove that f is (T, V )-continuous, i.e.,
⊤V ⩽ f◦
· a · Tf, which is equivalent to a(Tf(x), f(i)) = ⊤V for every x ∈ T2 and every i ∈ {0, 1}.
Since x ⩽ y implies k ⩽ a(eX(x), y) = a(eX(f(0)), y) = a((eX · f)(0), f(1)) = (diagram (1.1)) =
a((Tf · e2)(0), f(1)) = a(Tf(e2(0)), f(1)), and, similarly, y ⩽ x implies k ⩽ a(Tf(e2(1)), f(0)), one gets
a(Tf(e2(0)), f(1)) = ⊤V and a(Tf(e2(1)), f(0)) = ⊤V , since V is strictly two-sided.
Moreover, since (X, a) is a (T, V )-space, k ⩽ a(eX(x), x) = a(Tf(e2(0)), f(0)) and k ⩽ a(eX(y), y) =
a(Tf(e2(1)), f(1)) imply a(Tf(e2(0)), f(0)) = ⊤V and a(Tf(e2(1)), f(1)) = ⊤V , since k = ⊤V .
Lastly, by the assumption, it follows that a(Tf(x), f(i)) = ⊤V for every x ∈ T2 such that x ̸∈
{e2(0), e2(1)} and every i ∈ 2, which finishes the proof of (T, V )-continuity of f.
(3) Given a mono-source S = ((X, a)
fi
−→ (Yi, bi))i∈I in (T, V )-Cat with the property that (Yi, ai) is a
separated (T, V )-space for every i ∈ I, since the forgetful functor (T, V )-Cat
U
−→ Set has a left adjoint
(see Lecture 2), it preserves mono-sources, and, therefore, US = (X
fi
−→ Yi)i∈I is a mono-source in Set.
If I = ∅, then the set X has at most one element (since US is a mono-source), i.e., (X, a) is separated.
If I ̸= ∅, then to show that (X, a) is separated, take x1, x2 ∈ X such that x1 ⩽ x2 and x2 ⩽ x1, i.e.,
k ⩽ a(eX(x1), x2) and k ⩽ a(eX(x2), x1). Given i ∈ I, since (X, a)
fi
−→ (Yi, bi) is a (T, V )-continuous
2
map, it follows that a ⩽ f◦
i · bi · Tfi, which implies k ⩽ a(eX(x1), x2) ⩽ (f◦
i · bi · Tfi)(eX(x1), x2) =
bi((Tfi · eX)(x1), fi(x2)) = (since 1Set
e
−→ T is a natural transformation, the diagram
X
fi

eX
// TX
T fi

Yi eYi
// TYi
(1.2)
commutes, i.e., Tfi · eX = eYi
· fi) = bi((eYi
· fi)(x1), fi(x2)) = bi(eYi
(fi(x1)), fi(x2)) and, similarly,
k ⩽ bi(eYi
(fi(x2)), fi(x1)). Thus, fi(x1) ⩽ fi(x2) and fi(x2) ⩽ fi(x1), which implies fi(x1) = fi(x2),
since (Yi, bi) is separated. As a consequence, fi(x1) = fi(x2) for every i ∈ I, which provides x1 = x2,
since US is a mono-source in Set, i.e., point-separating. □
Remark 6.
(1) Since the category (T, V )-Catsep is closed under mono-sources in the category (T, V )-Cat, (T, V )-Catsep
is a strongly epireflective subcategory of the category (T, V )-Cat (see Lecture 5).
(2) In the category Top, for a topological space (X, τ), the respective Topsep-reflection arrow is given by the
quotient map X
p
−→ X/∼, where the equivalence relation ∼ on X is defined by x ∼ y iff cl({x}) = cl({y}),
in which cl(S) is the closure of a set S. Moreover, the quotient topology of the T0-space X/∼ makes
the above map p both U-final and U-initial w.r.t. the forgetful functor Top
U
−→ Set (see Lecture 2).
Proposition 7. Given a V -relation TX
a
// X, the following are equivalent:
(1) a · ˆTa ⩽ a · mX;
(2) a · ˆTa · m◦
X ⩽ a.
Proof.
(1) ⇒ (2): a · ˆTa ⩽ a · mX implies a · ˆTa · m◦
X ⩽ a · mX · m◦
X ⩽ a, since mX · m◦
X ⩽ 1T X.
(2) ⇒ (1): a · ˆTa · m◦
X ⩽ a implies a · ˆTa ⩽ a · ˆTa · m◦
X · mX ⩽ a · mX, since 1T T X ⩽ m◦
X · mX. □
Theorem 8. Given a (T, V )-space (X, a), the quotient map X
p
−→ X/∼, induced by the equivalence relation
∼ on the set X defined by x ∼ y iff x ⩽ y and y ⩽ x, provides a (T, V )-Catsep-reflection arrow for (X, a),
when X/∼ is equipped with the (T, V )-space structure ˜a = p · a · (Tp)◦
, i.e., such that the following diagram
TX
_a

T(X/∼)
_ ˜a

(T p)◦
oo
X p
// X/∼
commutes. This structure makes p both U-final and U-initial w.r.t. the forgetful functor (T, V )-Cat
U
−→ Set.
Proof. One begins with the proofs of several inequalities used later on.
(1) Notice that p◦
· p ⩽ a · eX, since given x, y ∈ X, it follows that (p◦
· p)(x, y) = [z]∼∈X/∼ p(x, [z]∼) ⊗
p◦
([z]∼, y) = [z]∼∈X/∼ p(x, [z]∼)⊗p(y, [z]∼) =
k, p(x) = p(y)
⊥V , otherwise
=
k, x ∼ y
⊥V , otherwise
⩽(a·eX)(x, y),
since x ∼ y implies x ⩽ y, which gives k ⩽ a(eX(x), y) = (a · eX)(x, y). Given x ∈ X, one uses here the
notation [x]∼ for the equivalence class of x w.r.t. the equivalence relation ∼, i.e., the set {y ∈ X | x ∼ y}.
3
(2) Observe that 1X/∼ ⩽ ˜a · eX/∼, since ˜a · eX/∼ = p · a · (Tp)◦
· eX/∼ ⩾ (since 1Set
e
−→ T is a natural
transformation, the following diagram
X
p

eX
// TX
T p

X/∼ eX/∼
// T(X/∼)
(1.3)
commutes, i.e., eX/∼ · p = Tp · eX, which implies (Tp)◦
· eX/∼ · p · p◦
= (Tp)◦
· Tp · eX · p◦
, which gives
(since p is surjective, and, therefore, p · p◦
= 1X/∼) (Tp)◦
· eX/∼ = (Tp)◦
· Tp · eX · p◦
, which provides
(Tp)◦
·eX/∼ ⩾ eX ·p◦
, since (Tp)◦
·Tp ⩾ 1T X) ⩾ p·a·eX ·p◦
⩾ (a·eX ⩾ p◦
·p by item (1)) ⩾ p·p◦
·p·p◦
=
(p · p◦
= 1X/∼, since p is surjective) = 1X/∼ · 1X/∼ = 1X/∼.
(3) Notice that p◦
·˜a·Tp ⩽ a, since p◦
·˜a·Tp = p◦
·p·a·(Tp)◦
·Tp ⩽ (p◦
·p ⩽ a·eX) ⩽ a·eX ·a·(Tp)◦
·Tp ⩽
(properties of lax extensions of functors imply (Tp)◦
·Tp ⩽ ˆT(p◦
)· ˆTp ⩽ ˆT(p◦
·p) ⩽ ˆT(a·eX) by item (1))
⩽ a·eX ·a· ˆT(a·eX) = (Lecture 2) = a·eX ·a· ˆTa·TeX ⩽ (since (X, a) is a (T, V )-space, a· ˆTa ⩽ a·mX)
⩽ a · eX · a · mX · TeX = (since T is a monad, mX · TeX = 1T X) = a · eX · a ⩽ (eX · a ⩽ ˆTa · eT X by a
property of lax extensions of monads) ⩽ a · ˆTa · eT X ⩽ (since (X, a) is a (T, V )-space, a · ˆTa ⩽ a · mX)
⩽ a · mX · eT X = (since T is a monad, mX · eT X = 1T X) = a.
(4) Observe that ˜a· ˆT ˜a ⩽ ˜a·mX/∼, since ˜a· ˆT ˜a·m◦
X/∼ = (since p is surjective, p·p◦
= 1X/∼, and, moreover,
the same holds for Tp and TTp, since Set-functors preserve surjective maps) = p · p◦
· ˜a · Tp · (Tp)◦
·
ˆT ˜a · TTp · (TTp)◦
· m◦
X/∼ ⩽ (p◦
· ˜a · Tp ⩽ a by item (3)) ⩽ p · a · (Tp)◦
· ˆT ˜a · TTp · (TTp)◦
· m◦
X/∼ ⩽
((Tp)◦
· ˆT ˜a · TTp ⩽ ˆT(p◦
) · ˆT ˜a · ˆTTp ⩽ ˆT(p◦
· ˜a · Tp) by the properties of lax extensions of functors of
Lecture 2) ⩽ p · a · ˆT(p◦
· ˜a · Tp) · (TTp)◦
· m◦
X/∼ = p · a · ˆT(p◦
· ˜a · Tp) · (mX/∼ · TTp)◦
= (since TT
m
−→ T
is a natural transformation, the following diagram
TTX
T T p

mX
// TX
T p

TTX/∼ mX/∼
// T(X/∼)
commutes, i.e., mX/∼ ·TTp = Tp·mX) = p·a· ˆT(p◦
·˜a·Tp)·(Tp·mX)◦
= p·a· ˆT(p◦
·˜a·Tp)·m◦
X ·(Tp)◦
⩽
(p◦
· ˜a · Tp ⩽ a by item (3)) ⩽ p · a · ˆTa · m◦
X · (Tp)◦
⩽ ((X, a) is a (T, V )-space backed by Proposition 7)
⩽ p · a · (Tp)◦
= ˜a, i.e., ˜a · ˆT ˜a · m◦
X/∼ ⩽ ˜a, which implies ˜a · ˆT ˜a ⩽ ˜a · mX/∼ by Proposition 7.
By the above items (2) and (4), (X, ˜a) is a (T, V )-space. Moreover, ˜a provides an U-final (T, V )-space
structure on the set X/∼ w.r.t. the map U(X, a)
p
−→ X/∼ (see Lecture 2). To show that a provides an
U-initial (T, V )-space structure on the set X w.r.t. the map X
p
−→ U(X/∼, ˜a), it is enough to check that
a = p◦
· ˜a · Tp (see Lecture 2). Notice that p◦
· ˜a · Tp ⩽ a by the above item (3). Moreover, ˜a = p · a · (Tp)◦
implies p◦
· ˜a · Tp = p◦
· p · a · (Tp)◦
· Tp ⩾ a, since p◦
· p ⩾ 1X as well as (Tp)◦
· Tp ⩾ 1T X.
To show that (X/∼, ˜a) is separated, notice that [x]∼ ⩽˜a [y]∼ implies p(x) ⩽˜a p(y), which gives k ⩽
˜a(eX/∼(p(x)), p(y)) = (p◦
· ˜a · eX/∼ · p)(x, y) = (diagram (1.3)) = (p◦
· ˜a · Tp · eX)(x, y) = (a · eX)(x, y) =
a(eX(x), y), i.e., x ⩽a y. If also [y]∼ ⩽˜a [x]∼, then, similarly, y ⩽a x, and, therefore, x ∼ y, i.e., [x]∼ = [y]∼.
To show that (X, a)
p
−→ (X/∼, ˜a) provides a (T, V )-Catsep-reflection arrow for (X, a), one has to check
that given a (T, V )-continuous map (X, a)
f
−→ (Y, b) with (Y, b) separated, there exists a unique (T, V )continuous
map (X/∼, ˜a)
˜f
−→ (Y, b), which makes the following triangle commute
(X, a)
p
//
f
%%
(X/∼, ˜a)
˜f

(Y, b).
(1.4)
4
Define the required map X/∼
˜f
−→ Y by ˜f([x]∼) = f(x). To show that the definition of the map ˜f
is correct, one has to check that [x1]∼ = [x2]∼ implies f(x1) = f(x2). Indeed, [x1]∼ = [x2]∼ implies
p(x1) = p(x2), which gives x1 ⩽a x2 and x2 ⩽a x1 by the previous paragraph. Thus, k ⩽ a(eX(x1), x2) ⩽
((X, a)
f
−→ (Y, b) is a (T, V )-continuous map) ⩽ b(Tf(eX(x1)), f(x2)) = b((Tf · eX)(x1), f(x2)) = (1Set
e
−→ T
is a natural transformation) = b((eY · f)(x1), f(x2)) = b(eY (f(x1)), f(x2)), i.e., f(x1) ⩽b f(x2), and,
similarly, f(x2) ⩽b f(x1). Since (Y, b) is a separated (T, V )-space, it follows that f(x1) = f(x2).
Commutativity of diagram (1.4) follows from the definition of the map ˜f. Moreover, since p is surjective,
the map ˜f, making diagram (1.4) commute, is unique. Lastly, since the map (X, a)
p
−→ (X/∼, ˜a) is U-final,
commutativity of diagram (1.4) implies that (X/∼, ˜a)
˜f
−→ (Y, b) is (T, V )-continuous (Lecture 2). □
Remark 9. Recall from Lecture 2 that there exists a concrete functor (T, V )-Cat
Spec
−−−→ Prost, which is
defined by Spec ((X, a)
f
−→ (Y, b)) = (X, ⩽a)
f
−→ (Y, ⩽b). The functor Spec restricts to the subcategories
(T, V )-Catsep of separated (T, V )-spaces and Prostsep = Pos of posets.
Corollary 10. The diagram
(T, V )-Catsep _
⊣

Spec
// Pos _
⊣

(T, V )-Cat
OO
Spec
// Prost
OO (1.5)
commutes w.r.t. both the solid and the dotted arrows.
Proof. Follows from the construction of (T, V )-Catsep-reflection arrows in Theorem 8. □
2. Between order separation and Hausdorff separation
Definition 11. A topological space (X, τ) is called
(1) T1-space provided that for every distinct x, y ∈ X, there exists U ∈ τ such that x ∈ U and y ̸∈ U;
(2) R0-space or symmetric space provided that for every x, y ∈ X, if x ∈ cl({y}), then y ∈ cl({x});
(3) R1-space provided that for every distinct x, y ∈ X, if cl({x}) ̸= cl({y}), then there exists U, V ∈ τ such
that x ∈ U, y ∈ V and U V = ∅.
Definition 12. Given a (T, V )-space (X, a), one can introduce the following separation axioms:
(T0) (a · eX) ∧ (a · eX)◦
⩽ 1X; (R0) (a · eX)◦
⩽ a · eX;
(T1) a · eX ⩽ 1X; (R1) a · a◦
⩽ a · eX.
Remark 13.
(1) Given a V -space (X, a), the axioms of Definition 12 simplify to the following:
(T0) a ∧ a◦
⩽ 1X; (R0) a◦
⩽ a;
(T1) a ⩽ 1X; (R1) a · a◦
⩽ a.
(2) The axioms of Definition 12 are inspired by the separation properties of topological spaces in the category
Top ∼= (β, 2)-Cat mentioned in Definitions 3, 11.
Lemma 14. For every (T, V )-space (X, a), eX ⩽ a◦
.
Proof. Since (X, a) is a (T, V )-space, it follows that 1X ⩽ a · eX, which implies 1X = (1X)◦
⩽ (a · eX)◦
=
e◦
X · a◦
, which provides eX ⩽ eX · e◦
X · a◦
⩽ a◦
, since eX · e◦
X ⩽ 1T X. □
5
Proposition 15. For every (T, V )-space (X, a), the following implications hold.
Hausdorff ⇔ (T1) & (R1)
⇓ ⇓ ⇓
(T1) ⇔ (T0) & (R0)
⇓
separated
Proof.
“Hausdorff ⇒ (T1) & (R1)”: Lemma 14 provides eX ⩽ a◦
, which implies a · eX ⩽ a · a◦
⩽ ((X, a) is
Hausdorff) ⩽ 1X ⩽ ((X, a) is a (T, V )-space) ⩽ a · eX. It follows that a · eX ⩽ 1X, which implies (T1).
Additionally, a · a◦
⩽ a · eX, which implies (R1).
“(T1) & (R1) ⇒ Hausdorff”: a · a◦
(R1)
⩽ a · eX
(T1)
⩽ 1X, i.e., a · a◦
⩽ 1X, i.e., (X, a) is Hausdorff.
“(T1) ⇒ (T0) & (R0)”: (a · eX) ∧ (a · eX)◦
⩽ a · eX
(T1)
⩽ 1X gives (X, a) is (T0); and a · eX
(T1)
⩽ 1X gives
(a · eX)◦
⩽ (1X)◦
= 1X ⩽ ((X, a) is a (T, V )-category) ⩽ a · eX, i.e., (a · eX)◦
⩽ a · eX, i.e., (X, a) is (R0).
“(T0) & (R0) ⇒ (T1)”: a · eX = (a · eX)◦◦
(R0)
⩽ (a · eX)◦
implies a · eX = (a · eX) ∧ (a · eX)◦
(T0)
⩽ 1X, i.e.,
a · eX ⩽ 1X, which implies (X, a) is (T1).
“(R1) ⇒ (R0)”: (a·eX)◦
= e◦
X ·a◦
⩽ (Lemma 14 gives eX ⩽ a◦
, which implies e◦
X ⩽ a) ⩽ a·a◦
(R1)
⩽ a·eX,
i.e., (a · eX)◦
⩽ a · eX, which implies that (X, a) is (R0).
“(T0) ⇒ separated”: Given x, y ∈ X such that x ⩽ y and y ⩽ x, it follows that k ⩽ a(eX(x), y) and
k ⩽ a(eX(y), x), which implies k ⩽ a(eX(x), y) ∧ a(eX(y), x) = (a · eX)(x, y) ∧ (a ·eX)(y, x) = (a ·eX)(x, y) ∧
(a · eX)◦
(x, y) = ((a · eX) ∧ (a · eX)◦
)(x, y)
(T0)
⩽ 1X(x, y), which gives x = y. Thus, (X, a) is separated. □
Corollary 16. For every V -space (X, a), the following implications hold.
Hausdorff ⇔ (T1) & (R1)
⇕ ⇓ ⇕
(T1) ⇔ (T0) & (R0)
⇓
separated
Moreover,
(1) (R0) is equivalent to a = a◦
;
(2) (T1) is equivalent to a = 1X;
(3) if V = 2, then “separated” implies (T0).
Proof. In view of Proposition 15, one shows just the additional implications and statements.
“(T1) ⇒ Hausdorff”: a
(T1)
⩽ 1X implies a◦
⩽ (1X)◦
= 1X implies a·a◦
⩽ a
(T1)
⩽ 1X, i.e., a·a◦
⩽ 1X, which
implies that (X, a) is Hausdorff.
“(R0) ⇒ (R1)”: a◦
(R0)
⩽ a implies a · a◦
⩽ a · a ⩽ ((X, a) is a V -category) ⩽ a, i.e., a · a◦
⩽ a, which
implies that (X, a) is (R1).
“(R0) ⇔ a = a◦
”: The sufficiency is clear. For the necessity, a◦
(R0)
⩽ a implies a = a◦◦
⩽ a◦
, i.e., a = a◦
.
“(T1) ⇔ a = 1X”: a
(T1)
⩽ 1X and 1X ⩽ a ((X, a) is a V -space) imply a = 1X.
“separated ⇒ (T0)”: By the assumption, V = 2 = ({⊥, ⊤}, ∧, ⊤). Given x, y ∈ X, it follows that
(a ∧ a◦
)(x, y) = a(x, y) ∧ a◦
(x, y) = a(x, y) ∧ a(y, x) = ⊤ iff a(x, y) = ⊤ and ⊤ = a(y, x) iff x ⩽ y and y ⩽ x
iff x
(X,a) is separated
= y iff 1X(x, y) = ⊤. As a consequence, one gets a ∧ a◦
⩽ 1X. □
Example 17.
6
(1) In the category 2-Cat ∼= Prost, (T0) coincides with the separation axiom of Definition 2 (2) by Corollary
16, i.e., both make posets from preordered sets. Moreover, (R0) coincides with (R1) by Corollary 16,
i.e., both make a preordered set (X, ⩽) symmetric (i.e., for every x, y ∈ X, x ⩽ y implies y ⩽ x), which
implies that the preorder ⩽ is an equivalence relation on X. Lastly, Hausdorffness and (T1) coincide
and make an equality relation “=” from a preorder “⩽” (see Lecture 5).
(2) In the category P+-Cat ∼= QPMet, (R0) coincides with (R1) by Corollary 16 and makes a quasi-pseudometric
space (X, ρ) symmetric, i.e., ρ(x, y) = ρ(y, x) for every x, y ∈ X. If (X, ρ) is symmetric, then
even (T0) makes ρ = 1X (see Corollary 16), i.e.,
ρ(x1, x2) =
0, x1 = x2
∞, otherwise,
and is, thus, considerably stronger than being order separated. However, a two-element quasi-pseudometric
space (X = {0, 1}, ρ) such that
ρ(x1, x2) =
∞, x1 = 0 and x2 = 1
0, otherwise,
is (T0) but not (T1), since ρ(1, 0) = 0 ̸= ∞.
(3) In the category Top ∼= (β, 2)-Cat, the axioms of Definition 12 are equivalent to their classical analogues
of general topology, which are mentioned in Definitions 3, 11.
(4) In the category App ∼= (β, P+)-Cat, one has the following straightforward characterizations:
• (X, a) is (T0) provided that for every x, y ∈ X, if a( ˙x, y) < ∞ and a( ˙y, x) < ∞, then x = y;
• (X, a) is (T1) provided that for every x, y ∈ X, if a( ˙x, y) < ∞, then x = y;
• (X, a) is (R0) provided that for every x, y ∈ X, a( ˙x, y) = a( ˙y, x);
• (X, a) is (R1) provided that for every x, y ∈ X and every z ∈ βX, a( ˙x, y) ⩽ a(z, x) + a(z, y), which
is equivalent to δ(y, {x}) ⩽ a(z, x)+a(z, y), where X ×PX
δ
−→ [0, ∞] is the approach distance of the
P+-category (X, a), defined by δ(z, C) = inf{a(y, z) | y ∈ βC} for every z ∈ X and every C ⊆ X.
Proposition 18. Given a topological construct C, if E is the class of C-bimorphisms (i.e., C-morphisms
which are both monomorphisms and epimorphisms), and M is the conglomerate of initial sources in C, then
(E, M) is a factorization system for sources in C.
Proposition 19.
(1) (T0) and (T1) separation properties are closed under mono-sources in (T, V )-Cat. Thus, the corresponding
full subcategories are strongly epireflective in (T, V )-Cat.
(2) (R0) and (R1) properties are closed under U-initial sources in (T, V )-Cat for the forgetful functor
(T, V )-Cat
U
−→Set. Thus, the respective full subcategories are both mono- and epireflective in (T, V )-Cat.
Proof.
(1) Take a mono-source S = ((X, a)
fi
−→ (Yi, bi))i∈I in (T, V )-Cat. Since the forgetful functor (T, V )-Cat
U
−→
Set has a left adjoint (see Lecture 2), it preserves mono-sources, i.e., US = (X
fi
−→ Yi)i∈I is a monosource
in Set. By the results of Lecture 5, it then follows that i∈I f◦
i · fi = 1X.
If (Yi, bi) is (T0) for every i ∈ I, then (a · eX) ∧ (a · eX)◦
⩽ ((X, a)
fi
−→ (Yi, bi) is a (T, V )-continuous map
for every i ∈ I) ⩽ i,j∈I(f◦
i · bi · Tfi · eX) ∧ (f◦
j · bj · Tfj · eX)◦ diagram (1.2)
= i,j∈I(f◦
i · bi · eYi
· fi) ∧ (f◦
j ·
bj · eYj
· fj)◦
⩽ i∈I(f◦
i · bi · eYi
· fi) ∧ (f◦
i · bi · eYi
· fi)◦
= i∈I(f◦
i · bi · eYi
· fi) ∧ (f◦
i · (bi · eYi
)◦
· fi) =
i∈I f◦
i · ((bi · eYi
) ∧ (bi · eYi
)◦
) · fi
(Yi,bi) is (T0)
⩽ i∈I f◦
i · fi = 1X, i.e., (a · eX) ∧ (a · eX)◦
⩽ 1X, which
then implies that the (T, V )-space (X, a) is (T0).
7
If (Yi, bi) is (T1) for every i ∈ I, then a · eX ⩽ ((X, a)
fi
−→ (Yi, bi) is a (T, V )-continuous map for every
i ∈ I) ⩽ i∈I f◦
i · bi · Tfi · eX
diagram (1.2)
= i∈I f◦
i · bi · eYi
· fi
(Yi,bi) is (T1)
⩽ i∈I f◦
i · fi = 1X, i.e.,
(a · eX) ∧ (a · eX)◦
⩽ 1X, which implies that (X, a) is (T1).
The last statement follows from the results of Lecture 5 on reflective subcategories.
(2) Given an U-initial source ((X, a)
fi
−→ (Yi, bi))i∈I in (T, V )-Cat, a = i∈I f◦
i · ai · Tfi by Lecture 2.
If (Yi, bi) is (R0) for every i ∈ I, then (a·eX)◦
⩽ ((X, a)
fi
−→ (Yi, bi) is a (T, V )-continuous map for every
i ∈ I) ⩽ i∈I(f◦
i · bi · Tfi · eX)◦ diagram (1.2)
= i∈I(f◦
i · bi · eYi · fi)◦
= i∈I f◦
i · (bi · eYi )◦
· fi
(Yi,bi) is (R0)
⩽
i∈I f◦
i ·bi ·eYi ·fi
diagram (1.2)
= i∈I f◦
i ·bi ·Tfi ·eX = ( i∈I f◦
i ·bi ·Tfi)·eX = a·eX, i.e., (a·eX)◦
⩽ a·eX,
which then implies that (X, a) is (R0).
If (Yi, bi) is (R1) for every i ∈ I, then a·a◦
⩽ ((X, a)
fi
−→ (Yi, bi) is a (T, V )-continuous map for every i ∈ I)
⩽ i∈I(f◦
i ·bi ·Tfi)· j∈I(f◦
j ·bj ·Tfj)◦
⩽ i∈I(f◦
i ·bi ·Tfi)·(f◦
i ·bi ·Tfi)◦
= i∈I f◦
i ·bi ·Tfi ·(Tfi)◦
·b◦
i ·fi ⩽
(Tfi ·(Tfi)◦
⩽ 1T Yi
) ⩽ i∈I f◦
i ·bi ·b◦
i ·fi
(Yi,bi) is (R1)
⩽ i∈I f◦
i ·bi ·eYi
·fi
diagram (1.2)
= i∈I f◦
i ·bi ·Tfi ·eX =
( i∈I f◦
i · bi · Tfi) · eX = a · eX, i.e., a · a◦
⩽ a · eX, which implies that (X, a) is (R1).
The last claim follows from the results of Lecture 5 on reflective subcategories and Proposition 18. □
3. Regular (T, V )-spaces
Definition 20. A topological space (X, τ) is called regular provided that for every x ∈ X and every closed
subset A ⊆ X such that x ̸∈ A, there exist U, V ∈ τ such that x ∈ U, A ⊆ V and U V = ∅.
Definition 21. A pair (X, a), where X is a set and TX a
// X is a V -relation is said to be a (T, V )-graph
provided that a is reflexive, i.e.,
X
1X
⩽
!!
eX
// TX
_ a

X.
Lemma 22. For a lax extension ˆT = ( ˆT, m, e) to V -Rel of a Set-monad T = (T, e, m), ˆT1X = ˆT(e◦
X)·m◦
X.
Proposition 23. Given a (T, V )-space (X, a), define TX ˆa
// TX = TX m◦
X
// TTX ˆT a
// TX. It
then follows that (TX, ˆa) is a V -graph, but (X, a · ˆa) and (X, a · ˆa◦
) are (T, V )-graphs. Moreover, a · ˆa ⩽ a
is equivalent to the transitivity condition for a.
Proof. Notice that 1T X = T1X ⩽ (properties of lax extensions of functors) ⩽ ˆT1X = (Lemma 22)
= ˆT(e◦
X) · m◦
X ⩽ (Lemma 14) ⩽ ˆTa · m◦
X = ˆa, i.e., 1T X ⩽ ˆa, which implies that (TX, ˆa) is a V -graph.
Further, a·ˆa·eX ⩾ ((TX, ˆa) is a V -graph) ⩾ a·eX ⩾ ((X, a) is a (T, V )-space) ⩾ 1X, i.e., a·ˆa·eX ⩾ 1X,
which implies that (X, a · ˆa) is a (T, V )-graph.
Lastly, a · ˆa◦
· eX ⩾ ((TX, ˆa) is a V -graph) ⩾ a · (1T X)◦
· eX = a · 1T X · eX = a · eX ⩾ ((X, a) is a
(T, V )-space) ⩾ 1X, i.e., a · ˆa◦
· eX ⩾ 1X, which implies that (X, a · ˆa◦
) is a (T, V )-graph.
Lastly, a · ˆa ⩽ a iff a · ˆTa · m◦
X ⩽ a iff a · ˆTa ⩽ a · mX by Proposition 7. □
Definition 24.
(1) A (T, V )-space (X, a) is called regular provided that a · ˆa◦
⩽ a, i.e., a · mX · ( ˆTa)◦
⩽ a, or, in pointwise
notation, ˆTa(Y, x) ⊗ a(mX(Y), x) ⩽ a(x, x) for every Y ∈ TTX, x ∈ TX, and every x ∈ X.
(2) The full subcategory of (T, V )-Cat of regular spaces is denoted (T, V )-Catreg.
8
Proposition 25. A V -category (X, a) is regular iff a = a◦
.
Proof. Given a V -category (X, a), it follows that ˆa = a. Thus, regularity is equivalent to a · a◦
⩽ a, which
is exactly (R1). By Corollary 16, (R1) is equivalent to a = a◦
. □
Example 26.
(1) In the categories 2-Cat ∼= Prost and P+-Cat ∼= QPMet, by Proposition 25, preordered sets and
quasi-pseudo-metric spaces are regular exactly when they are symmetric (recall Example 17).
(2) In the category Top ∼= (β, 2)-Cat, regular topological spaces are precisely the regular spaces in the sense
of general topology of Definition 20.
(3) In the category App ∼= (β, P+)-Cat, an approach space (X, a) is regular precisely when for every x, y ∈
βX and every x ∈ X, it follows that a(x, x) ⩽ ˆa(y, x) + a(y, x), where ˆa(y, x) = inf{u ∈ [0, ∞] | A(u)
∈
x for every A ∈ y} (recall Lecture 1 for the notation A(u)
).
Proposition 27.
(1) If V is lean and strictly two-sided, and ˆT is flat, then every compact Hausdorff (T, V )-space is regular.
(2) The subcategory (T, V )-Catreg is closed in the category (T, V )-Cat under U-initial sources for the forgetful
functor (T, V )-Cat
U
−→ Set, and, therefore, is both mono- and epireflective in (T, V )-Cat.
Proof.
(1) Given a compact Hausdorff (T, V )-space (X, a), if V is a lean and strictly two-sided quantale, then the
V -relation TX a
// X is a map, and, moreover, it follows that a · Ta = a · mX (see Lecture 5). It
then follows that a · ˆa◦
= a · ( ˆTa · m◦
X)◦
= a · mX · ( ˆTa)◦ T is flat
= a · mX · (Ta)◦ a·mX =a·T a
= a · Ta · (Ta)◦
⩽
(Ta · (Ta)◦
⩽ 1T X) ⩽ a, i.e., a · ˆa◦
⩽ a, i.e., (X, a) is regular.
(2) Given an U-initial source ((X, a)
fi
−→ (Yi, bi))i∈I in (T, V )-Cat, a = i∈I f◦
i · bi · Tfi by Lecture 2.
If (Yi, bi) is regular for every i ∈ I, then for every i ∈ I, it follows that a · ˆa◦
= a · ( ˆTa · m◦
X)◦
= a · mX ·
( ˆTa)◦
⩽ ((X, a)
fi
−→ (Yi, bi) is a (T, V )-continuous map) ⩽ f◦
i ·bi ·Tfi ·mX ·( ˆT(f◦
i ·bi ·Tfi))◦
= (for every
map X
f
−→ Y and every V -relations Y s
// Z, Z r
// Y, ˆT(s·f) = ˆTs·Tf and ˆT(f◦
·r) = (Tf)◦
· ˆTr
by Lecture 2) = f◦
i · bi · Tfi · mX · ((Tfi)◦
· ˆTbi · TTfi)◦
= f◦
i · bi · Tfi · mX · (TTfi)◦
· ( ˆTbi)◦
· Tfi =
(since TT
m
−→ T is a natural transformation, the following diagram
TTX
T T fi

mX
// TX
T fi

TTYi mYi
// TYi
commutes, i.e., Tfi·mX = mYi
·TTfi) = f◦
i ·bi·mYi
·TTfi·(TTfi)◦
·( ˆTbi)◦
·Tfi ⩽ (TTfi·(TTfi)◦
⩽ 1T T Yi
)
⩽ f◦
i · bi · mYi · ( ˆTbi)◦
· Tfi
(Yi,bi) is regular
⩽ f◦
i · bi · Tfi. As a consequence, it follows that a · ˆa◦
⩽
i∈I f◦
i · bi · Tfi = a, i.e., a · ˆa◦
⩽ a, which implies that (X, a) is regular.
The last claim follows from the results of Lecture 5 on reflective subcategories and Proposition 18. □
Remark 28. A regular (T, V )-space may not be Hausdorff (or even separated). This can be seen, e.g., for
V -spaces: Hausdorffness means discreteness (Lecture 5), and regularity means symmetry (Proposition 25).
Definition 29. Given a functor Set
T
−→ Set, a lax extension V -Rel
ˆT
−→ V -Rel of T to V -Rel is said to be
symmetric provided that ˆT(r◦
) = ( ˆTr)◦
for every V -relation X r
// Y.
9
Proposition 30. Given a morphism of symmetric lax extensions of monads ˆS
α
−→ ˆT, the respective algebraic
functor (T, V )-Cat
Aα
−−→ (S, V )-Cat, Aα((X, a)
f
−→ (Y, b)) = (X, a · αX)
f
−→ (Y, b · αY ) preserves regularity.
Proof. Suppose that S = (S, n, d) and take a regular (T, V )-space (X, a). To show that the (S, V )space
(X, a · αX) is regular, notice that a · αX · a · αX
◦
= a · αX · ( ˆS(a · αX) · n◦
X)◦
= (for every map
X
f
−→ Y and every V -relation Y s
// Z, ˆS(s · f) = ˆSs · Sf by Lecture 2) = a · αX · ( ˆSa · SαX · n◦
X)◦
=
a · αX · nX · (SαX)◦
· ( ˆSa)◦
ˆS is symmetric
= a · αX · nX · (SαX)◦
· ˆS(a◦
) = (by Lecture 2, since S
α
−→ T is a
morphism of monads, the following diagram
SS
n

α◦α // TT
m

S α
// T
commutes, where α ◦ α is defined by the diagonal of the commutative diagram
SS
αS
 ""
Sα // ST
αT

TS
T α
// TT,
(3.1)
i.e., α ◦ α = Tα · αS = αT · Sα) = a · mX · TαX · αSX · (SαX)◦
· ˆS(a◦
) ⩽ (diagram (3.1) implies TαX · αSX =
αT X · SαX, which gives αSX · (SαX)◦
⩽ (TαX)◦
· TαX · αSX · (SαX)◦
= (TαX)◦
· αT X · SαX · (SαX)◦
⩽
(TαX)◦
·αT X, since 1T SX ⩽ (TαX)◦
·TαX and SαX ·(SαX)◦
⩽ 1ST X) ⩽ a·mX ·TαX ·(TαX)◦
·αT X · ˆS(a◦
) ⩽
(TαX · (TαX)◦
⩽ 1T T X) ⩽ a · mX · αT X · ˆS(a◦
) ⩽ (since α is a morphism of lax extensions of functors,
SX
αX
//
_ˆS(a◦
)

⩽
TX
_ ˆT (a◦
)

STX αT X
// TTX,
i.e., αT X · ˆS(a◦
) ⩽ ˆT(a◦
) · αX) ⩽ a · mX · ˆT(a◦
) · αX
ˆT is symmetric
= a · mX · ( ˆTa)◦
· αX = a · ( ˆTa · m◦
X)◦
· αX =
a · ˆa◦
· αX
(X,a) is regular
⩽ a · αX, i.e., a · αX · a · αX
◦
⩽ a · αX. □
Remark 31. Given a lax extension ˆT of a monad T = (T, m, e) on Set, I
e
−→ ˆT is a morphism of lax
extensions of monads, where I = (1Set, 1, 1) is the identity monad on Set.
Corollary 32. Given a symmetric lax extension ˆT of a monad T = (T, m, e) on Set, the algebraic functor
(T, V )-Cat
Ae
−−→ V -Cat, Aα((X, a)
f
−→ (Y, b)) = (X, a · eX)
f
−→ (Y, b · eY ) preserves regularity.
Proof. The claim follows from Remark 31 and Proposition 30. □
Remark 33. If (T, V ) = (β, P+) (the symmetricity condition is satisfied for the lax extension of β to P+-Rel
of Lecture 1), then Corollary 32 says that the underlying metric of a regular approach space is symmetric.
4. Normal and extremally disconnected (T, V )-spaces
Definition 34.
10
(1) A topological space (X, τ) is said to be normal provided that for every disjoint closed subsets A, B ⊆ X,
there exist disjoint elements U, V ∈ τ such that A ⊆ U and B ⊆ V .
(2) A topological space (X, τ) is extremally disconnected if the closure of every open subset of X is open.
Proposition 35. For every topological space (X, τ) represented as a (β, 2)-space (X, a), equivalent are:
(1) (X, τ) is a normal topological space;
(2) ˆa · ˆa◦
⩽ ˆa◦
· ˆa.
Definition 36. A (T, V )-space (X, a) is called normal provided that ˆa·ˆa◦
⩽ ˆa◦
·ˆa, or, in pointwise notation,
a(x, y) ⊗ a(x, z) ⩽ s∈T X a(y, s) ⊗ a(z, s) for every x, y, z ∈ TX.
Definition 37. A lax extension ˆT to V -Rel of a monad T = (T, m, e) on Set is associative provided that
t · ˆT(s · ˆTr · m◦
X) · m◦
X = t · ˆTs · m◦
Y · ˆTr · m◦
X for all unitary V -relations TX r
// Y, TY s
// Z, and
TZ
t
// W, where a V -relation TX
r
// Y is unitary provided that r · ˆT1X ⩽ r and e◦
Y · ˆTr · m◦
X ⩽ r.
Proposition 38. For every lax extension ˆT to V -Rel of a monad T = (T, m, e) on Set, equivalent are:
(1) ˆT is associative;
(2) V -Rel
ˆT
−→ V -Rel preserves composition and ˆT
m◦
−−→ ˆT ˆT is natural.
Proposition 39. If ˆT is associative, then for every (T, V )-space (X, a), equivalent are:
(1) (X, a) is normal;
(2) (TX, ˆa) is a normal V -space;
(3) (TX, ˆa◦
· ˆa) is a V -space.
Proof.
“(1) ⇔ (2)”: Notice that given a V -space (Y, b), it follows that ˆb = b. Thus, (TX, ˆa) is a normal V -space
iff ˆa · ˆa◦
⩽ ˆa◦
· ˆa iff (X, a) is a normal (T, V )-space. It remains to show that if (X, a) is a (T, V )-space, then
(TX, ˆa) is a V -space. By Proposition 23, (TX, ˆa) is a V -graph, i.e., 1T X ⩽ ˆa, which proves reflexivity. To
show transitivity, notice that ˆa · ˆa = ˆTa · m◦
X · ˆTa · m◦
X = (since ˆT is associative, ˆT
m◦
−−→ ˆT ˆT is natural by
Proposition 38, and, therefore, the following diagram
TTX 
m◦
T X
//
_ˆT a

TTTX
_ ˆT ˆT a

TX 
m◦
X
// TTX,
commutes, i.e., m◦
X · ˆTa = ˆT ˆTa·m◦
T X) = ˆTa· ˆT ˆTa·m◦
T X ·m◦
X = ˆTa· ˆT ˆTa·(mX ·mT X)◦
= (since T = (T, m, e)
is a monad, the following diagram
TTT
mT

T m // TT
m

TT m
// T
commutes, i.e., mX · mT X = mX · TmX) = ˆTa · ˆT ˆTa · (mX · TmX)◦
= ˆTa · ˆT ˆTa · (TmX)◦
· m◦
X ⩽
((TmX)◦
⩽ ˆT(m◦
X) by the properties of lax extensions of monads of Lecture 2) ⩽ ˆTa · ˆT ˆTa · ˆT(m◦
X) · m◦
X ⩽
(properties of lax extensions of monads of Lecture 2) ⩽ ˆT(a· ˆTa·m◦
X)·m◦
X ⩽ (since (X, a) is a (T, V )-category,
a · ˆTa ⩽ a · mX, which implies a · ˆTa · m◦
X ⩽ a by Proposition 7) ⩽ ˆTa · m◦
X = ˆa.
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“(2) ⇒ (3)”: By Proposition 23, (TX, ˆa) is a V -graph, i.e., 1T X ⩽ ˆa, which implies 1T X = (1T X)◦
⩽ ˆa◦
,
and, therefore, 1T X = 1T X · 1T X ⩽ ˆa◦
· ˆa, which proves reflexivity. To show transitivity, notice that
ˆa◦
· ˆa · ˆa◦
· ˆa
(X,a) is normal
⩽ ˆa◦
· ˆa◦
· ˆa · ˆa = (ˆa · ˆa)◦
· ˆa · ˆa
(X,a) is a V -space
⩽ ˆa◦
· ˆa.
“(3) ⇒ (1)”: ˆa · ˆa◦
⩽ ((TX, ˆa◦
· ˆa) is a V -space) ⩽ ˆa · ˆa◦
· ˆa · ˆa◦
⩽ ((TX, ˆa) is a V -graph implies 1T X ⩽ ˆa
implies 1T X ⩽ ˆa◦
) ⩽ ˆa◦
· ˆa, i.e., ˆa · ˆa◦
⩽ ˆa◦
· ˆa, which proves normality of (X, a). □
Proposition 40. For every topological space (X, τ) represented as a (β, 2)-space (X, a), equivalent are:
(1) (X, τ) is extremally disconnected;
(2) ˆa◦
· ˆa ⩽ ˆa · ˆa◦
.
Definition 41. A (T, V )-space is called extremally disconnected provided that ˆa◦
· ˆa ⩽ ˆa · ˆa◦
.
Proposition 42. A V -space (X, a) is normal iff (X, a◦
) is extremally disconnected.
Proof. Given a V -space (X, a), (X, a◦
) is a V -space by the results of Lecture 4. Moreover, since ˆa = a,
(X, a) is normal iff a · a◦
⩽ a◦
· a iff (a◦
)◦
· a◦
⩽ a◦
· (a◦
)◦
iff (X, a◦
) is extremally disconnected. □
Proposition 43. If ˆT is associative, then for every (T, V )-space (X, a), equivalent are:
(1) (X, a) is extremally disconnected;
(2) (TX, ˆa) is an extremally disconnected V -space;
(3) (TX, ˆa◦
) is a normal V -space;
(4) (TX, ˆa · ˆa◦
) is a V -space.
Proof.
“(1) ⇔ (2)”: See the respective item of the proof of Proposition 39.
“(2) ⇔ (3)”: Follows from Proposition 42.
“(3) ⇔ (4)”: Follows from “(2) ⇔ (3)” of Proposition 39. □
Definition 44. Given a preordered set (X, ⩽), the preorder ⩽ is said to be confluent provided that for
every x, y, z ∈ X, if x ⩽ y and x ⩽ z, then there exists s ∈ X such that y ⩽ s and z ⩽ s. Co-confluence is
defined dually: for every x, y, z ∈ X, if y ⩽ x and z ⩽ x, then there is s ∈ X such that s ⩽ y and s ⩽ z.
Remark 45. Given a preordered set (X, ⩽), if the preorder ⩽ is symmetric (i.e., for every x, y ∈ X, if
x ⩽ y, then y ⩽ x), then ⩽ is both confluent and co-confluent.
Example 46.
(1) A V -space (X, a) is normal iff for every x, y, z ∈ X, it follows that a(x, y)⊗a(x, z) ⩽ s∈X a(y, s)⊗a(z, s).
Moreover, (X, a) is extremally disconnected iff for every x, y, z ∈ X, it follows that a(y, x) ⊗ a(z, x) ⩽
s∈X a(s, y) ⊗ a(s, z). In particular, a preordered set (X, ⩽) considered as a 2-category is normal iff
the preorder ⩽ is confluent. Moreover, (X, ⩽) is extremally disconnected iff the preorder ⩽ is coconfluent.
Thus, a normal (T, V )-space is not necessarily regular. However, a regular V -space (X, a),
i.e., a symmetric V -space (a = a◦
by Proposition 25), is both normal and extremally disconnected.
(2) A topological space considered as a (β, 2)-category (X, a) is normal or extremally disconnected iff it is
normal or extremally disconnected in the sense of general topology (Propositions 35, 40). Moreover, by
Proposition 39, (X, a) is normal iff the preorder ⪯ (equal to ˆa) on βX is confluent.
(3) In the category QPMet ∼= P+-Cat, a quasi-pseudo-metric space (X, a) is normal iff for every x, y, z ∈ X,
it follows that a(x, y) + a(x, z) ⩾ infs∈X a(y, s) + a(z, s).
(4) In the category App ∼= (β, P+)-Cat, an approach space considered as a (β, P+)-space (X, a) is normal
iff for every x, y, z ∈ βX, it follows that ˆa(x, y)+ˆa(x, z) ⩾ infs∈βX ˆa(y, s)+ˆa(z, s), where ˆa(x, y) = inf{u ∈
[0, ∞] | A(u)
∈ y for every A ∈ x} (recall Lecture 1 for the notation A(u)
).
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Proposition 47. If ˆT is associative and flat, then every T-algebra is a normal (T, V )-space.
Proof. Given a T-algebra (X, a), by Proposition 39, it is enough to show that (TX, ˆa) is a normal V -space.
Since a is a map TX
a
−→ X, for every x, y ∈ TX, it follows that
ˆa(x, y) = ( ˆTa · m◦
X)(x, y)
T is flat
= (Ta · m◦
X)(x, y) =
Z∈T T X
m◦
X(x, Z) ⊗ Ta(Z, y) =
Z∈T T X
mX(Z, x) ⊗ Ta(Z, y) =
k, ∃Z ∈ TTX : mX(Z) = x, Ta(Z) = y
⊥V , otherwise,
(4.1)
i.e., ˆa is completely determined by its induced preorder ⩽ on TX of Definition 2. Thus, to show that (TX, ˆa)
is a normal V -space, by Example 46 (1), one has to verify that the induced preorder ⩽ on TX is confluent.
Given x, y, z ∈ TX such that x ⩽ y and x ⩽ z, in view of formula (4.1), there exist Y, Z ∈ TTX such that
mX(Y) = x = mX(Z) and Ta(Y) = y, Ta(Z) = z. Since (X, a) is a T-algebra, one obtains a · Ta = a · mX,
and, therefore, for y = a(y) and z = a(z), it follows that y = a(y) = a(Ta(Y)) = a · Ta(Y) = a · mX(Y) =
a(x) = a · mX(Z) = a · Ta(Z) = a(Ta(Z)) = a(z) = z. We now show that y ⩽ eX(y) and z ⩽ eX(z), which
will finish the proof, since y = z implies eX(y) = eX(z).
For y ⩽ eX(y), notice that for W = eT X(y), mX(W) = mX(eT X(y)) = mX ·eT X(y) = (m·eT = 1T , since
T is a monad) = y and Ta(W) = Ta(eT X(y)) = Ta · eT X(y) = (since 1Set
e
−→ T is a natural transformation,
the following diagram
TX
a

eT X
// TTX
T a

X eX
// TX
commutes, i.e., Ta · eT X = eX · a) = eX · a(y) = eX(a(y)) = eX(y). The case z ⩽ eX(z) is similar. □
Corollary 48. If the quantale V is strictly two-sided and lean, and ˆT is associative and flat, then every
compact Hausdorff (T, V )-space is normal.
Proof. The claim follows from Proposition 47 and the fact that if V is a strictly two-sided and lean
quantale, and ˆT is flat, then (T, V )-CatCompHaus = SetT
(see Lecture 5). □
References
[1] J. Ad´amek, H. Herrlich, and G. E. Strecker, Abstract and Concrete Categories: the Joy of Cats, Repr. Theory Appl. Categ.
17 (2006), 1–507.
[2] A. S. Davis, Indexed systems of neighborhoods for general topological spaces, Am. Math. Mon. 68 (1961), 886–894.
[3] R. Engelking, General Topology, Sigma Series in Pure Mathematics, vol. 6, Heldermann Verlag, 1989.
[4] H. Herrlich, Topologie I. Topologische R¨aume. Unter Mitarb. von H. Bargenda, vol. 2, Heldermann Verlag, Lemgo, 1986.
[5] D. Hofmann, G. J. Seal, and W. Tholen (eds.), Monoidal Topology: A Categorical Approach to Order, Metric and Topology,
Cambridge University Press, 2014.
[6] R. Lowen, Index Analysis. Approach Theory at Work, London: Springer, 2015.
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