with respect to their splittability behaviour. The interested reader is referred to [6, 11, 12, 13, 14]

SPLITTABILITY AND LOW SEPARATION AXIOMS

A.J. Hanna and T.B.M. McMaster

(Received August 2003)

Abstract. The power and usefulness of cleavability (also known as splittabil- ity) have been well established within the framework of topology by A.V.

Arhangel’skii and his associates. The concept was transferred to partially or- dered sets by D.J. Marron and T.B.M. McMaster. The connections between topology and order are here exploited as we examine the properties enjoyed by a topological space that is splittable over a collection of topological spaces each enjoying a low separation axiom.

1. Introduction

Given the close links between topology and order it is hardly surprising that the definition and basic results of splittability (for example, see [2, 3, 4]) transfer naturally from topological spaces to partially ordered sets [7, 8, 9, 10]. This article examines a selection of separation axioms weaker than T

1

with respect to their splittability behaviour. The interested reader is referred to [6, 11, 12, 13, 14]

for more details on specific properties. In each case we determine, using a variety of techniques, whether the invariant is a splittability class or not: that is, we test whether splittability over a class of P spaces implies membership of P.

In some cases the ‘full force’ of splittability is not required in order to establish such a conclusion. For example, we sometimes find that pointwise splittability over a class of P spaces suffices to imply membership of P. For other invariants we find it necessary to strengthen the type of splittability used. Failing this, we need to insist on some further condition on the ‘split’ space coupled with the splitting property.

2. Preliminary Material

Definition 2.1. A topological space X is splittable [2] over a class P of topological spaces if for every subset A of X there exists a continuous map f : X → Y ∈ P such that f (A) ∩ f (X \ A) = ∅. We use the terms closed (open, onto) splittable to mean that the splitting maps may be chosen to be closed (open, onto). The space X is said to be pointwise splittable over P if splitting maps can be found at least for every singleton subset of X. When working with posets we need only replace the term ‘continuous map’ by that of ‘increasing map’ i.e. a map f for which x ≤ y in X implies f (x) ≤ f (y)
. A class of spaces P is said to be a splittability class if X splittable over P implies X ∈ P. It is easy to see that splittability classes must be expansive and hereditary.

1991 Mathematics Subject Classification Primary 06A06, 54C99.

Key words and phrases: splittable, submaximal, partially ordered set, low separation.

Definition 2.2. Let f : X → Y be a mapping between posets. We use the notations L(x) = {y ∈ X : y ≤ x} and M (x) = {y ∈ X : x ≤ y} and we say [8]

that:

(1) f is an L

P

map if f (L(x)) = L(f (x)) for all x ∈ X, (2) f is an M

_P

map if f M (x)
= M f (x)
for all x ∈ X.

It is easily shown that L

P

maps and M

P

maps are increasing.

It is well known that a T

0

topological space X generates a poset under the rela- tionship x ≤ y ⇐⇒ x ∈ {y}. Several topologies may be defined on a given poset that generate its order in this fashion. However, there is a weakest such topology:

Definition 2.3. Given a poset (X, ≤), the weak topology is the topology on X whose non-empty proper closed subsets are generated by the family {L(x) : x ∈ X}.

Thus we can consider the two point Sierpi´ nski space (denoted by C

2

) as the poset on {1, 2} (with natural order) endowed with the weak topology. Likewise we denote by C

n

the topological space obtained by imposing the weak topology on the set {1, 2, . . . , n} with its usual ordering. Throughout this paper all posets will be treated as the topological spaces formed by imposing the weak topology unless stated otherwise.

Theorem 2.4. Let (X, τ ), (Y, τ

⁰

) be finite T

₀

topological spaces and let (X, ≤

_X

), (Y, ≤

_Y

) be their corresponding poset representations. Let f : X → Y be a map from X to Y ; then

(1) f is a continuous map if and only if f is an increasing map, (2) f is an open continuous map if and only if f is an M

P

map, and (3) f is a closed continuous map if and only if f is an L

_P

map.

Note also that all three left–to–right implications are valid without the assumption of finiteness here.

The following result of Marron [7] will prove invaluable throughout this paper.

Theorem 2.5. Let X be a finite poset; then X is splittable over C

n

if and only if X contains neither an n + 1 element chain nor two disjoint n element chains.

Definition 2.6. Let (X, τ ) be a topological space. Then:

• X ∈ T

₁

if and only if, for all x ∈ X, {x} is τ –closed.

• X ∈ T

EF

if and only if every finite subset of X is either τ –closed or τ –open.

• X ∈ T

ES

if and only if, for all x ∈ X, {x} is either τ –closed or τ –open.

• X ∈ T

D

if and only if, for all x ∈ X, {x}\{x} is τ –closed.

• X ∈ T

0

if and only if, for all x, y ∈ X, {x} = {y} implies x = y.

• X ∈ T

F

if and only if, given a finite subset F of X and x / ∈ F , there exists a τ –neighbourhood of one which excludes the other.

• X ∈ T

F F

if and only if, given two disjoint finite subsets F

1

, F

2

of X, there exists a τ –neighbourhood of one which excludes the other.

• X ∈ T

F D

if and only if X is both T

F

and T

D

.

The results gathered together in the next theorem may all be deduced from their definitions in a routine manner.

Theorem 2.7. Each of the separation axioms described above is a splittability class.

With the exception of T

EF

and T

F F

, pointwise splittability over each invariant is sufficient to imply that invariant. Note in particular that T

F D

is a splittability class since it is the intersection of the two splittability classes T

F

and T

D

.

Definition 2.8. Let (X, τ ) be a topological space. Then:

• X ∈ T

U D

if and only if, for all x ∈ X, either {x} is τ –closed or {x}\{x} is a union of disjoint τ –closed sets.

• X ∈ T

A

if and only if, for all x ∈ X, either {x} is τ –closed or {x} is τ –open or {x}\{x} is a point closure.

• X ∈ T

F A

if and only if X is both T

F

and T

A

.

• X ∈ T

SD

if and only if, for all x ∈ X, either {x} is τ –closed or {x}\{x} = {y}

where {y} is τ –closed.

• X ∈ T

SA

if and only if, for all x ∈ X, either {x} is τ –closed or {x} is τ -open or {x}\{x} = {y} where {y} is τ –closed.

• X ∈ T

δ

if and only if, for all x ∈ X, either {x} is τ –closed or {x} \ {x} is a point closure.

• X ∈ T

ζ

if and only if, for all x ∈ X, either {x} is τ –closed or {x}\{x} is a union of incommensurable point closures. (A non-empty family F of subsets of X is incommensurable when for each distinct pair F

₁

, F

₂

∈ F , F

₁

* F

2

and F

₂

* F

1

.)

• X ∈ T

Y

if and only if, for all x 6= y in X,{x} ∩ {y} has at most one element.

• X ∈ T

_{Y Y}

if and only if there exists p ∈ X such that for all x 6= y in X, {x} ∩ {y} = {x}, {y}, {p} or ∅.

• X ∈ T

Y S

if and only if, for all x 6= y in X, {x} ∩ {y} = {x}, {y} or ∅.

• X ∈ T

_DD

if and only if X is both T

_{Y S}

and T

_D

.

• X ∈ door if and only if, for all A ⊆ X, A is either τ –closed or τ –open.

• X ∈ submaximal if and only if, for all A ⊆ X, A is locally τ –closed (i.e. the intersection of a τ –closed set and a τ –open set). Equivalently, X is submaximal if and only if every dense subset of X is τ –open.

3. Results

Proofs of the earlier results in this section are straightforward and have, apart from a couple of samples, been omitted. The first is due to A. V. Arhangel’skiˇı [3]

and shows in effect that door is a splittability class.

Theorem 3.1. A space is door if and only if it is splittable over the Sierpi´ nski space.

Theorem 3.2. A space is submaximal if and only if it splittable over a class of

submaximal spaces. Thus, submaximal is a splittability class.

Theorem 3.3. Let Y

SM

be the space shown below. A space is submaximal if and only if it is splittable over Y

SM

.

i 1

i 3

i2 i4

@

@

@ @

Suppose that T

α

and T

β

are separation axioms such that T

β

implies T

α

. If T

β

fails to be a splittability class then ‘X is splittable over T

β

’ does not imply X ∈ T

β

. How can we best strengthen the condition to restore the result? Two methods are apparent; either restrict the class of mapping (for example ‘X is closed splittable over T

β

’) or allow X to be nearly T

β

(for example ‘X is T

α

and splittable over T

β

’).

Example 3.4. Let X be the space shown below.

i b

ia ic A

A A A

Define maps f

1

, f

2

, f

3

: X → C

2

as follows:

f

₁

(a) = 0 f

₁

(b) = 1 f

₁

(c) = 1, f

₂

(a) = 0 f

₂

(b) = 0 f

₂

(c) = 1, f

3

(a) = 0 f

3

(b) = 1 f

3

(c) = 0.

Clearly these maps are increasing and split along each subset of X. Recall that X is not T

Y S

(since {b} ∩ {c} = {a}) while C

2

is T

Y S

, so we have found a non-T

_{Y S}

space splittable over the class of T

_{Y S}

spaces. That is, T

_{Y S}

is not a splittability class. (Alternatively, if T

_{Y S}

is a splittability class then door implies T

_{Y S}

– a contradiction.)

Note that the splittability of X over C

₂

follows directly from Theorem 2.5.

However, the maps f

₁

, f

₂

, f

₃

are actually L

_P

(therefore closed) and X is already T

Y Y

; hence even the stronger condition ‘X is T

Y Y

and closed splittable over the class of T

Y S

spaces’ does not imply that X is T

Y S

. We can also see that ‘X is closed splittable over the class of T

DD

spaces’ does not imply that X is a T

DD

space.

Lemma 3.5 ([6]). A space X is T

F

if and only if, for all x ∈ X, y ∈ {x}\{x}

implies {y} is closed.

Theorem 3.6. A space is T

_{Y S}

if and only if it is open (pointwise) splittable over a class of T

Y S

spaces.

Corollary 3.7. A space is T

_DD

if and only if it is open (pointwise) splittable over a class of T

_DD

spaces.

Example 3.8. Let X be the space shown below; then X is neither T

δ

, nor T

A

, nor

T

SA

. Clearly C

3

is T

δ

, T

A

and T

SA

. It can be shown that X is M

P

splittable over

C

3

so the condition ‘X is T

D

and open splittable over the class of T

δ

(T

A

, T

SA

)

spaces’ does not imply that X is a T

δ

(T

A

, T

SA

) space. Indeed we see that if X is

T

_D

and open splittable over a class of T

_SA

spaces then X may fail even to be T

_A

.

i

y iz

ix iw

A A

A A

X

i1 i2 i3 C

3

i

u

3

iv

3

ix

4

ix

5

ix

1

ix

2

A A

A A

A A

A A

Y

iy

1

iy

2

iy

3

iy

4

iy

5

C

5

Example 3.9. Let Y, C

5

be the spaces shown above. We note that C

5

is T

A

while Y is not: because {x

₄

} is not closed nor is {x

4

} \ {x

4

} a point closure. Consider the three maps f : Y → C

₅

, g : Y → C

₅

, h : Y → C

₅

which take x

_i

to y

_i

for i = 1, 2, 4 and 5 and whose definitions are completed as follows: f (u

₃

) = f (v

₃

) = y

₃

, g(u

₃

) = y

₃

and g(v

₃

) = y

₂

, h(u

₃

) = y

₂

and h(v

₃

) = y

₃

. These are continuous and closed, and they split Y along every one of its subsets over C

₅

. Moreover, Y and C

5

are self–dual, so we can readily perceive open continuous maps that also suffice to split. Hence ‘Y is T

D

and both closed splittable and open splittable over T

A

’ fails to imply that Y is T

A

.

Theorem 3.10. A space is T

SA

if and only if it is T

A

and is splittable over a class of T

SA

spaces.

Theorem 3.11. A space is T

SA

if and only if it is closed splittable over a class of T

_SA

spaces.

Corollary 3.12. A space is T

_{F A}

if and only if it is closed splittable over a class of T

_{F A}

spaces.

Example 3.13. Let X be the dual of the space used in Example 12. Dual argu- ments to those used there show that X is open splittable over C

2

. Now X is not T

_SD

(since {a} is neither closed nor is {a}\{a} a closed singleton) while C

2

is T

_SD

. Hence even the stronger condition ‘X is T

_{F A}

and T

_Y

and open splittable over the class of T

_SD

spaces’ does not imply that X is a T

_SD

space.

Theorem 3.14. A space is T

_SD

if and only if it is closed (pointwise) splittable over a class of T

SD

spaces.

Theorem 3.15. A space is T

U D

if and only if it is (pointwise) splittable over a class of T

U D

spaces. Thus, T

U D

is a splittability class.

Theorem 3.16. A space is T

Y

if and only if it is splittable over a class of T

Y

spaces. Thus, T

Y

is a splittability class.

Proof. Clearly every T

Y

space is splittable over itself. Suppose that X / ∈ T

Y

but that X is splittable over a class P of T

Y

spaces. There must exist distinct x, y ∈ X such that {x} ∩ {y} is neither empty nor a singleton. We note first that by Theorem 2.7, X is T

F

so x / ∈ {y} and y / ∈ {x} via Lemma 3.5. Hence there must be distinct u, v ∈ {x} ∩ {y}. Consider the subspace Y = {u, v, x, y}. Since Y is finite and T

_F

, every singleton is either open or closed, so Y must have the topology τ = {∅, Y, {x}, {y}, {x, y}, {x, y, u}, {x, y, v}} (any stronger topology would fail to satisfy u, v ∈ {x} ∩ {y}). Note that this is just Y

SM

as defined in Theorem 3.3.

Since X is splittable over P, so is Y . Choose continuous f : Y → Z ∈ P such that f

⁻¹

f ({x, u}) = {x, u}. Now f (Y ) is finite T

Y

and hence T

ES

.

• If |f (Y )| = 2 then f ({x, u}) is a singleton so {x, u} is open or closed in X.

• If |f (Y )| = 3 then f ({x, u}) or f ({y, v}) is a singleton so either {x, u} or {y, v}

is open in X.

• If |f (Y )| = 4 then f (Y ) cannot be T

Y

since f is continuous and T

_Y

is expansive.

We have reached a contradiction, hence no such X can exist.  Example 3.17. Let X be the space defined on {x

i

, y

i

, z

i

: i ∈ N} with y

ⁱ

, z

i

< x

i

for all i ∈ N as shown below; then X is T

^F

, but not T

A

since {x

1

} is neither open, nor closed, nor is {x

1

}\{x

1

} a point closure.

i

y

₁

iz

1

ix

1

A A

A A

y

₂

i iz

2

ix

2

A A

A A

y

₃

i iz

3

ix

3

A A

A A

y

₄

i iz

4

ix

4

A A

A A

...

Let Y be the space defined on {a

_i

, b

_i

: i ∈ N} with b

i

< a

_i

for all i ∈ N as shown below; then Y is T

_F

and T

_A

.

ib

1

ia

1

ib

2

ia

2

ib

3

ia

3

ib

4

ia

4

...

Given A ⊆ X define a map f : X → Y as follows:

If A ∩ {x

_i

, y

_i

, z

_i

} = {z

i

} or {x

i

, y

_i

} let f (x

i

) = a

_i

, f (y

_i

) = a

_i

, f (z

_i

) = b

_i

. If A ∩ {x

_i

, y

_i

, z

_i

} = {y

i

} or {x

i

, z

_i

} let f (x

i

) = a

_i

, f (y

_i

) = b

_i

, f (z

_i

) = a

_i

. Otherwise let f (x

_i

) = a

_i

, f (y

_i

) = b

_i

, f (z

_i

) = b

_i

.

Now f is continuous and f

⁻¹

f (A) = A so X is splittable over Y . Moreover f is an open map since f (X \ {z

_i

}) and f (X \ {y

_i

}) equal Y or Y \ {b

_i

} while f (X \ {y

i

, z

i

}) equals Y \ {b

i

}. This shows that a space X may be T

F D

and open splittable over T

SD

without even being T

A

.

Example 3.18. Let X = {x, y, p, u, v, q} and Y = {a, b, c, d} be the spaces shown

below.

i

x iy

ip

A A A A

i

u iv

iq

A A A A X

ic ia

id ib Y

Examining Example 3.4 we see that X is closed splittable over Y . We note that Y is T

_{Y Y}

while X is not because {x} ∩ {y} = {p} 6= {q} = {u} ∩ {v}. Hence T

_{Y Y}

is not a splittability class. Moreover, even the stronger condition ‘X is T

_Y

and closed splittable over the class of T

_{Y S}

spaces’ does not imply that X is a T

_{Y Y}

space.

Lemma 3.19. A space X is T

Y Y

if and only if X is open splittable over a class P of T

Y Y

spaces and X has no six–point subspace with induced order as shown below.

i

x iy

ip

A A A A

i

w iz

iq

A A A A

Theorem 3.20. Let (X, ≤) be a partially ordered set, upon which is induced either the weak or the Alexandroff topology; then the space X is T

Y Y

if and only if X is open splittable over a class of T

Y Y

spaces.

Proof. Again we need only prove sufficiency. Let X be open splittable over a class P of T

_{Y Y}

spaces. We note that (X, ≤) has height less than or equal to 2. It is clear that if X has no subspace as depicted and labelled in Lemma 3.19 then X is Y

_{Y Y}

. We therefore assume that X contains this subspace.

Let B = {u ∈ X : u ∈ {v} \ {v} for some v ∈ X \ {x, w}}. Choose open continuous f : X → Y ∈ P that splits along A = {x, p, w} ∪ B. Clearly

f (p) ∈ {f (x)} ∩ {f (y)}, and f (q) ∈ {f (w)} ∩ {f (z)}.

 (∗)

Consider G = X \ ({w} ∪ {x}) which is open in X. Clearly z ∈ G so f (z) ∈ f (G).

If f (w) ∈ f (G) then there exists a ∈ G such that f (a) = f (w) ∈ f (A). Since a / ∈ {w} ∪ {x} we must have a ∈ B. There exists b / ∈ A such that a ∈ {b} \ {b}

so f (a) ∈ {f (b)}. However, f (q) ∈ {f (w)} = {f (a)} contradicting Y ∈ T

Y Y

, so f (w) / ∈ f (G). It follows that f (z) / ∈ {f (w)} so f (z) 6= f (q). For condition (∗) to hold in the T

_{Y Y}

space Y we must have f (x) = f (p). If X has the Alexandroff topology then {x} is an increasing set in (X, ≤) and must be open. The set {f (x)}

is therefore open so f (p) = f (x) / ∈ {f (y)}, contradicting condition (∗). If X has

the weak topology we consider two separate cases.

Case 1. If f

⁻¹

f (y) is finite we label its points {a

1

, a

2

, . . . , a

n

}. Set G = X \ ({a

1

} ∪ {a

2

} ∪ . . . ∪ {a

n

} ∪ {y}).

Now x ∈ G so f (x) ∈ f (G). If f (y) ∈ f (G) then there exists b ∈ G such that f (y) = f (b) so b = a

i

for some i ≤ n. However, b = a

i

∈ G – a contradiction. / Therefore f (y) / ∈ f (G) so f (p) = f (x) / ∈ {f (y)} contradicting condition (∗).

Case 2. If f

⁻¹

f (y) is infinite then the set f

⁻¹

({f (y)}) is both infinite and closed.

If f

⁻¹

({f (y)}) = X then f (w) ∈ {f (y)}. Hence f (q), f (w) ∈ {f (u)} ∩ {f (y)}, contradicting the fact that Y ∈ T

Y Y

. There are also infinitely many u ∈ f

⁻¹

f (y) that are not members of B and therefore not contained in any point derived set.

It follows that f

⁻¹

({f (y)}) cannot be the union of finitely many point closures, a contradiction to X having the weak topology.

 We deal next with the axiom T

ζ

, which turns out to be tractible by the methods shown above but somewhat less easily. The strong link between partially ordered sets and topologies that are at least T

0

is particularly valuable here, allowing us to characterize this and similar axioms in a convenient and intuitive fashion. For instance, in [13] we find the following definition: a topological space (X, τ ) is T

ζ

if and only if the induced order is a partial order such that, given x < y in X, then y has a pseudo-predecessor x such that x ≤ z. Here, a pseudo–predecessor of y is a maximal element of L(y)\{y}.
A straightforward application of transfinite induction will show that a partially ordered set E that has no maximal element can be decomposed into the disjoint union of two sets, each of which is cofinal in E.

Splitting along one of these two sets provides a way to focus on maximality issues, which we now exploit.

Theorem 3.21. A space is T

ζ

if and only if it is closed splittable over a class of T

ζ

spaces.

Proof. As usual, the necessity of the condition is immediate. To establish the sufficiency, suppose if possible that X is not T

_ζ

but is splittable by closed continuous mappings over T

_ζ

spaces. Because X is at least T

₀

, its induced order is a partial order, and in terms of it we may select x ∈ X such that the set E = {a ∈ X : a <

x but no pseudo–predecessor ofxlies abovea} is not empty. Express E as a disjoint union A ∪ B of sets, each cofinal in E, and select a

₀

∈ A. Take P to denote the set (possibly empty) of all pseudo–predecessors of x and find a closed continuous mapping f from X to a T

ζ

space Y that splits along A ∪ P . Note (see Theorem 2.4) that f is increasing and an L

p

map.

Every point e of E can be made the first term of an increasing sequence e <

a

1

< b

1

< a

2

< b

2

< a

3

< . . . that alternates between A and B, and whose images under the splitting map f must all be distinct. This shows, firstly, that f (a

0

) is strictly less than f (x), and so f (x) has a pseudo–predecessor m for which f (a

0

) ≤ m < f (x). Now m ∈ L f (x)
= f L(x)
, so we may choose y ≤ x such that m = f (y). Clearly y < x and, by a second use of the above observation, y cannot belong to E. Thus x has a pseudo-predecessor P satisfying y ≤ p < x.

Again, f (a

₀

) ∈ L(m) = L f (y)
= f L(y)
, so select z ≤ y for which f (z) = f (a

0

).

The splitting condition assures us that z ∈ A ∪ P but, since z ≤ y ≤ p ∈ P, this is only possible when z = y = p. But now f (a

0

) = f (z) = f (y) = m, which is contradicted by a third appeal to the above. This concludes the proof.  The separation axioms T

A

and T

δ

require special treatment. The techniques previously employed to establish pleasing results break down. Instead we adopt a different approach by defining a weakened form of each axiom. This property is then bestowed on spaces to be split over a class of spaces with the original property. We begin with T

δ

. We have already seen in previous examples that open splittability and closed splittability over a class of T

δ

spaces fail to imply T

δ

. Below we define a weakened form of T

δ

and show that a space with this weakened property that is closed splittable over T

δ

must itself be T

δ

.

Definition 3.22. A space X is nearly T

δ

if and only if, for all x ∈ X, either {x} is closed or {x} \ {x} is a point closure or there exist A, B ⊆ X such that A ∩ B = ∅, {x}\{x} = A ∪ B, A ⊆ B and B ⊆ A. It is easily shown that T

^δ

⇒ nearly T

δ

, but that nearly T

δ

does not even imply T

0

.

Theorem 3.23. A space is T

_δ

if and only if it is nearly T

_δ

and closed splittable over a class of T

_δ

spaces.

Proof. We need only prove sufficiency. Suppose that X / ∈ T

δ

but X is nearly T

δ

and closed splittable over a class P of T

δ

spaces. There must exist x ∈ X so that {x} is not closed nor is {x}\{x} a point closure. Now there exist A, B ⊆ X satisfying the conditions in the definition. Choose continuous closed f : X → Y ∈ P with f

⁻¹

f (A) = A. Then, firstly:

f (A) ⊆ f (B) ⊆ f ({x}) = {f (x)} and f (B) ⊆ f (A).

Now {f (x)} is not closed since A 6= ∅, so there exists p ∈ Y for which {f (x)}\{f (x)} = {p}. As in the previous proof we can find c ∈ {x}\{x} such that f (c) = p, and this c must lie either in A or in B. The splitting condition assures us that f (A) and f (B) are disjoint. If c ∈ A then p ∈ f (A)\f (B) so either f (x) ∈ f (B) or f (B) ⊆ {p}\{p}

which is of the form {q}. The first case gives f (x) ∈ f (B) ⊆ f (A) ⊆ {p} which is a contradiction, while the second yields p ∈ f (A) ⊆ f (B) ⊆ {q} which is another.

On the other hand, if c ∈ B a like pair of contradictions arise.  Remark 3.24. The order–theoretic description of T

_δ

, found in [13], permits an alternative view of the previous proof: a topology (X, τ ) is T

_δ

if and only if the induced order ≤ is a partial order such that every non–minimal element of X has a predecessor (that is, a maximum strict lower bound).

Thus far, this account has presented only negative results concerning the separa- tion axiom T

_A

. Perhaps surprising, we can obtain a positive result by introducing a weaker version of T

_A

analogous to nearly T

_ζ

.

Definition 3.25. A space X is nearly T

A

if and only if, for all x ∈ X, either {x}

is open or {x} is closed or {x}\{x} is a point closure or there exist A, B ⊆ X such

that A ∩ B = ∅, {x} \{x} = A ∪ B, A ⊆ B and B ⊆ A. It is again easily seen

that T

A

⇒ nearly T

A

, but that nearly T

A

does not even imply T

0

. The proof of the

following proceeds largely as that of Theorem 3.21.

Theorem 3.26. A space is T

A

if and only if it is nearly T

A

and closed splittable over a class of T

A

spaces.

References

[1] A.V. Arhangel’skiˇı, On cleavability of topological spaces over R, R

ⁿ

, and R

^ω

, Ann. New York Acad. Sci. 659 (1992), 18–28.

[2] A.V. Arhangel’skiˇı, The general concept of cleavability of a topological space, Top. Applic. 44 (1992), 27–36.

[3] A.V. Arhangel’skiˇı, A survey of cleavability, Top. Applic. 54 (1993), 141–163.

[4] A.V. Arhangel’skiˇı and F. Cammaroto, On different types of cleavability of topological spaces, J. Austral. Math. Soc. (Series A), 58 (1995), 183–199.

[5] A.V. Arhangel’skiˇı and B.E. Shapirovskiˇi, On splittable spaces, International conference on topology, Varna, (1990), 9–10.

[6] C.E. Aull and W.J. Thron, Separation axioms between T

0

and T

1

, Indag. Math.

24 (1962), 26–37.

[7] D.J. Marron, Splittability in ordered sets and in ordered spaces, PhD thesis, Queen’s University Belfast, 1997.

[8] D.J. Marron and T.B.M. McMaster, Splittability in ordered sets and spaces, Proc. Eighth Prague Topological Symp. (1996), 280–282.

[9] D.J. Marron and T.B.M. McMaster, Splittability for finite partially-ordered sets, Math. Proc. R. Ir. Acad. 99A (2) (1999), 189–194.

[10] D.J. Marron and T.B.M. McMaster, Splittability for ordered topological spaces, Boll. Un. Mat. Ital., 8 (1-B) (2000) 213–220.

[11] A.E. McCluskey and S.D. McCartan, The minimal structures for T

_A

, Ann.

New York Acad. Sci. 659 (1992), 138–155.

[12] A.E. McCluskey and S.D. McCartan, Minimality with respect to T

_SA

and T

_SD

, Topology with Applications, Szeksz´ ard (Hungary), (1993), 83–97.

[13] A.E. McCluskey and S.D. McCartan, Minimal structures for T

F A

, Rend. Ist.

Mat. Univ. Trieste, 27 (1995), 11–24.

[14] D. M. G. McSherry, On separation axioms weaker than T

1

, Proc. R. Ir. Acad.

74A (1974), 115–118.

A.J. Hanna

Queen’s University Belfast Belfast BT7 1NN

UNITED KINGDOM [email protected]

T.B.M. McMaster Queen’s University Belfast Belfast BT7 1NN

UNITED KINGDOM [email protected]