The Hierarchy of Borel Universal Sets

March 12, 2000

A $Σ_{α}^{0}$ -subset $U$ of the product $X \times Y$ is a $Σ_{α}^{0}$ -universal set of $X$ parametrised by $Y$ if every $Σ_{α}^{0}$ -set of $X$ is of the form $U^{y} = {x : (x, y) \in U}$ , for some $y \in Y$ . Let $n \in ω$ and $α \in ω_{1}$ . If $X$ is a compact space with a $Σ_{n}^{0}$ -universal set parametrised by $Y$ , then for all $m \in ω$ , $w (X) \leq n w (Y)$ , $h d (X^{m}) \leq h d (Y^{m})$ , $h L (X^{m}) \leq h L (Y^{m})$ and $h c (X^{m}) \leq h c (Y^{m})$ . If $X$ is a perfect compact space with a $Σ_{α}^{0}$ -universal set parametrised by $Y$ , then $w (X) \leq n w (Y)$ . There is an example of a space that is not second countable, but with a $G_{δ}$ -universal set parametrised by the Cantor set, $2^{ω}$ . Assuming $b = ω_{1}$ , there is a locally compact strong $S$ -space with a $G_{δ}$ -universal set parametrised by $2^{ω}$ . Assuming CH, there is an $L$ -space with a $G_{δ}$ -universal set parametrised by $2^{ω}$ . Assuming $b = ω_{1}$ , there is a compact strong $S$ -space with a $G_{δ}$ -universal parametrised by a strong $S$ -space. If there exist Q-sets or under CH, there is a compact, first countable non-metrisable space with a $Σ_{ω}^{0}$ -universal set parametrised by $2^{ω}$ . The statements ‘every compact monotonically normal space with a $Σ_{α}^{0}$ -universal set parametrised by a second countable space is metrisable’ and ‘every compact, first countable space with a $Σ_{α}^{0}$ -universal set parametrised by a second countable space is metrisable’ are undecidable in ZFC.

Keywords: Borel hierarchy, Borel universals, cardinal invariants, S and L spaces, compact spaces
MCS: 54H05, 54D30 Secondary 54D65, 54E35, 54D15

Introduction

Let $Γ$ be a class function assigning to each topological space a family $Γ (X)$ of subsets of $X$ . A set $U$ contained in $X \times Y$ is a $Γ$ -universal set for $X$ parametrised by $Y$ if $U \in Γ (X \times Y)$ and $Γ (X) = {U^{y} : y \in Y}$ , where $U^{y} = {x \in X : (x, y) \in U}$ . Universal sets have played an important role in the study of the Borel hierarchies and analytic sets ([8, Sections 14 and 22]). See also [9], [10] and [13]. Their work was mainly restricted to Polish spaces and parametrising countable closed sets, countable $G_{δ}$ -sets or compact sets. In the present paper, we investigate Borel universal sets of general regular Hausdorff spaces, relating properties of the space $X$ to that of the parametrising space $Y$ . We pay particular attention to the case when $X$ is compact. Section 1 provides definitions, notations and basic results.

The Borel sets of a space form a natural hierarchy of additive $Σ_{α}^{0}$ - and multiplicative $Π_{α}^{0}$ -classes $(0 \neq α \in ω_{1})$ . A space has a $Σ_{α}^{0}$ -universal set parametrised by a space $Y$ if and only if it has a $Π_{α}^{0}$ -universal set parametrised by $Y$ (Lemma 1). Thus we formulate results solely in terms of $Σ_{α}^{0}$ -universal sets. If a space has a $Σ_{α}^{0}$ -universal set parametrised by a space $Y$ , then it has $Σ_{β}^{0}$ -universal sets parametrised by $Y^{ω}$ for all $β \geq α$ (Proposition 6). Consequently, we have a hierarchy of results. We will see, however, that this collapses into three groups: open ( $Σ_{1}^{0}$ -) universal sets, $Σ_{n}^{0}$ -universal sets for $1 < n < ω$ (in other words, $F_{σ}$ -, $G_{δ σ}$ - etc sets), and $Σ_{α}^{0}$ -universal sets for $α \geq ω$ .

Every space has a $Σ_{α}^{0}$ -universal set parametrised by a metrisable space, and one parametrised by a compact space (for each $α$ ) (Lemma 8). The question, then, is which spaces have Borel universal sets parametrised by `small’ and `nice’ spaces? Suppose, for the moment that $2^{ℵ_{0}} < 2^{ℵ_{1}}$ and $X$ is a space with a $Σ_{α}^{0}$ -universal set (any $α$ ) parametrised by a space of size no more than the continuum. As the discrete space of size $ℵ_{1}$ has $2^{ℵ_{1}}$ many $Σ_{α}^{0}$ -sets, cardinality considerations alone ensure that $X$ must have the hereditary c.c.c. On the other hand, if $2^{ℵ_{0}} = 2^{ℵ_{1}}$ , then the discrete space of size $ℵ_{1}$ has an open universal set parametrised by a compact, first countable, separable space. (See Theorem 10 for details.) Two points flow immediately from this observation. First, since space with the hereditary c.c.c. are either both hereditarily Lindelöf and hereditarily separable, or contain a hereditarily Lindelöf non-separable subspace, or contain a hereditarily separable non-Lindelöf subspace, we are inevitably led to the famous $S$ - and $L$ -space problems. Second, when looking for absolute theorems, our parametrising space will need to have a strong property hereditarily. Regarding the result on hereditary cellularity above, we give characterisations of spaces with a $Σ_{n}^{0}$ - $(0 < n < ω)$ or $Σ_{α}^{0}$ - $(α \geq ω)$ universal set parametrised by a separable metrisable space (Theorems 18 and 30). From this it follows that every space with a $Σ_{α}^{0}$ -universal set parametrised by a separable metrisable space satisfies the c.c.c. hereditarily if and only if there are no $Q_{\leq α}$ -sets (Theorem 32).

We summarise our other results according to the grouping mentioned above. This corresponds to Sections 2, 3 and 4 below. Each section is further divided in two, with the first part dealing with general spaces, and the second the special case of compact spaces. All results in Section 2 have been mentioned in [5], which is devoted to open universal sets.

Space $X$ with an open universal set parametrised by $Y$ :

Theorem 11 and Corollary 12 For any $m \in ω$ ,

\begin{matrix} w (X) & \leq n w (Y), h c (X^{m}) \leq h c (Y^{m}), \\ h d (X^{m}) & \leq h L (Y^{m}), and h L (X^{m}) \leq h d (Y^{m}) . \end{matrix}

In particular, a space is separable metrisable if and only if parametrised by a separable metrisable space. Note also the duality of hereditary density and the hereditary Lindelöf property.

Example 14 It is independent of ZFC, that $Y$ hereditarily separable implies $X$ hereditarily separable, and $Y$ hereditarily Lindelöf implies $X$ hereditarily Lindelöf.

When $X$ is compact the direct implications do hold.

Compact $X$ with an open universal set parametrised by $Y$ :

Lemma 16(2) and Corollary 17 The space $X$ is the continuous image of a subspace of $Y$ . Hence any hereditary property, preserved by continuous maps, is transferred from $Y^{m}$ to $X^{m}$ .

In particular, $h d (X^{m}) \leq h d (Y^{m})$ and $h L (X^{m}) \leq h L (Y^{m})$ .

Turning to general spaces with $Σ_{n}^{0}$ -universal sets $(1 < n < ω)$ , most of the open universal set results fail.

Space $X$ with a $Σ_{n}^{0}$ -universal set parametrised by $Y$ $(1 < n < ω)$ :

Example 19 There is a hereditarily separable and hereditarily Lindelöf, but non-metrisable, space, with an $F_{σ}$ -universal set parametrised by the Cantor set.

Examples 20 and 21 (CH) There are $S$ - and $L$ -spaces with $F_{σ}$ -universal sets parametrised by the Cantor set.

For compact $X$ , the situation is more positive.

Compact $X$ with a $Σ_{n}^{0}$ -universal set parametrised by $Y$ $(1 < n < ω)$ :

Corollaries 24 and 25 For $m \in ω$ ,

\begin{matrix} w (X) & \leq n w (Y), h d (X^{m}) \leq h d (Y^{m}), \\ h L (X^{m}) & \leq h L (Y^{m}), and h c (X^{m}) \leq h c (Y^{m}) . \end{matrix}

Example 27 ( $b = ω_{1}$ ) There is a compact $S$ -space with an $F_{σ}$ -universal set parametrised by a strong $S$ -space.

Finally, from $Σ_{ω}^{0}$ onwards, there are `bad’ compact spaces with Cantor parametrising space.

Compact $X$ with $Σ_{α}^{0}$ -universal set parametrised by $Y$ $(α \geq ω)$ :

Examples 34 and 35 If there is a $Q_{\leq ω}$ -set, or if (CH), then there is a compact non-metrisable space with a $Σ_{ω}^{0}$ -universal set parametrised by the Cantor set.

Placing additional restrictions on $X$ , we have the following:

Corollaries 36, 37, 38 Let $X$ be a compact space with a $Σ_{α}^{0}$ -universal set parametrised by a separable metrisable space. Then $X$ is metrisable if:

(1) $X$ is perfect, or

(2) (Consistently) $X$ is monotonically normal, or

(3) (Consistently) $X$ is first countable.

1 Definitions and useful results

All spaces are regular Hausdorff topological spaces unless stated otherwise. Our topological notation follows that of [3]. Definitions of the cardinal invariants used here can be found in [6] or [5]. An introduction to $S$ - and $L$ -spaces can be found in [14].

Borel sets and the Borel hierarchy: Let $X$ be a space. Then the Borel subsets of $X$ are those in the smallest $σ$ -algebra containing the open sets of $X$ . They ramify into the following hierarchy. Define $Σ_{1}^{0} (X)$ to be all open subsets of $X$ , and, inductively on $α \in ω_{1} - {0}$ , $Π_{α}^{0} (X) = {X - S : S \in Σ_{α}^{0}}$ , $Σ_{α + 1}^{0} (X)$ is all countable unions of members of $Π_{α}^{0} (X)$ , and $Σ_{α}^{0} (X)$ is the family of countable unions of members of $⋃_{β < α} Σ_{β}^{0} (X)$ for limit $α$ . Then the union of all the $Σ_{α}^{0}$ - and $Π_{α}^{0}$ -sets is precisely the family of all Borel sets. We define a subset of $X$ to be a $Σ_{α}^{0}$ -set if it is an element of $Σ_{α}^{0} (X)$ . Note that $Π_{1}^{0} (X)$ is the family of closed subsets of $X$ , and $Σ_{2}^{0} (X)$ are the $F_{σ}$ -subsets.

Borel universal sets: A $Σ_{α}^{0}$ -set $U$ of the space $X \times Y$ is a $Σ_{α}^{0}$ -universal set of $X$ parametrised by $Y$ if for each $A \in Σ_{α}^{0} (X)$ there is a $y \in Y$ such that $A = U^{y}$ , where $U^{y} = {x \in X : (x, y) \in U}$ .

Orders on $ω^{ω}$ : Define orders on $ω^{ω}$ as follows: $f <^{*} g$ if for all but finitely many $n$ , $f (n) < g (n)$ ; $f \leq g$ if for all $n$ , $f (n) \leq g (n)$ ; and $f \leq_{l e x} g$ if $f$ precedes $g$ in the lexicographic order. The cardinal $b$ is the supremum of all cardinals $κ$ such that there is a subset of $ω^{ω}$ of order type $κ$ under $<^{*}$ .

Subsets of $R$ : An uncountable subset of the real line is a $λ$ -set if every countable subset is a (relative) $G_{δ}$ . Identifying $ω^{ω}$ with the irrationals, any subset of $ω^{ω}$ of order type $ω_{1}$ under $<^{*}$ is a $λ$ -set. (Hence $λ$ -sets exist.) An uncountable subset of the reals is a $Q_{\leq α}$ -set if every subset is a (relative) $Σ_{α}^{0}$ -set. The existence of $Q_{\leq α}$ -sets is undetermined by the standard axioms of set theory. For all uncountable cardinal $κ$ , the existence of $Q_{\leq α}$ -sets of size $κ$ is undecidable in ZFC. (Existence is implied by MA+ $2^{ω} > κ$ , and non-existence by CH.) See [12] and [11] for more on subsets of the reals.

Todorcevic’s examples: There are many (consistent) examples of $S$ - and $L$ -spaces. We will make use of what may now be called `classical’ (CH) examples due to Kunen (the `Kunen line’) [14] and van Douwen et al. [1]. But we will also need more recent constructions of Todorcevic.

Let $X$ be any subset of the real line, and let $τ$ be a topology on $X$ finer than the Euclidean topology. We will call $(X, τ)$ a Kunen line type space, if every $τ$ -closure of a subset of $X$ differs from the Euclidean closure of $X$ only by countably many elements. It is easy to see that any Kunen line type space is hereditarily separable. The Kunen line is of Kunen line type (!), as is the Sorgenfrey topology on any subset of the reals.

Assume $b = ω_{1}$ , and fix a subset, $A$ , of $ω^{ω}$ of order type $ω_{1}$ under $<^{*}$ . To repeat, $A$ is a $λ$ -subset of $R$ . In chapter 2 of his book [16], Todorcevic associates to a certain type of map, $H$ , of $A$ into the countable subsets of $A$ a space $A [H]$ with the following properties: the topology on $A [H]$ is of Kunen line type, and $A [H]$ is a locally compact strong $S$ -space.

In chapter 0 of the same monograph, for any partial order $\leq$ on $A$ , Todorcevic defines the space $A [\leq]$ with basic open neighbourhoods of $a$ in $A$ of the form, for $n \in ω$ , $B_{n, a} [\leq a] = {b \in A : b \leq a} \cap {b \in A : a | n = b | n}$ . Clearly the topology on $A [\leq]$ refines the Euclidean topology. Todorcevic proves (Theorem 0.6) that $A [\leq]$ is a strong $S$ -space and $A [\geq]$ is a strong $L$ -space. He also shows (Theorem 3.0) that $A [\leq_{l e x}]$ and $A [\geq_{l e x}]$ (which are homeomorphic to subsets of the Sorgenfrey line) are hereditarily separable and hereditarily Lindelöf in all finite powers. In [5, Section 1.4], it was noted that $A [\leq]$ is also a Kunen line type space.

Duplication: Let $X$ be a compact subset of $R$ , let $τ$ be a topology on $X$ , finer than the Euclidean topology, with a base of compact open sets, and let $C$ be a $τ$ -closed subset of $X$ . Define $D (X, τ, C)$ to be the subspace $D (X, τ, X) - (X - C) \times {0}$ , of the space $D (X, τ, X)$ which has underlying set $X \times {0, 1}$ , and topology with base consisting of sets of the form $T \times {0}$ for $T$ in $τ$ , and $U \times {0, 1} - K \times {0}$ for Euclidean open $U$ and $K$ a compact-open subspace of $(X, τ)$ . Note that $D (I, D, I)$ , where $D$ is the discrete topology, is the usual Alexandroff duplicate.

Clearly $D (X, τ, C)$ is a closed subspace of $D (X, τ, X)$ . It is straightforward to check that $D (X, τ, X)$ is a compact (Hausdorff) space which is: first countable, separable, hereditarily separable or metrisable, if and only if $(X, τ)$ has the corresponding property.

Preservation of universal sets: From now on, $Γ$ will denote a Borel class. Write $\sim Γ (X)$ for ${X - S : S \in Γ (X)}$ . Most of the results of this section are modifications of those concerning the open universal sets found in [5, Section 1.6].

Lemma 1 If

U

is a

Γ

-universal set for

X

parametrised by

Y

, then

(X \times Y) - U

is a

\sim Γ

-universal set for

X

parametrised by

Y

Lemma 2 If

X^{'}

is a subspace of

X

, then

U \cap (X^{'} \times Y)

is a

Γ

-universal set for

X^{'}

parametrised by

Y

Proof. The result follows from the fact that the trace of a Borel set on a subspace is a Borel set of that subspace of the same Borel class. $◊$

Lemma 3 (1) If

Y^{'} \supseteq Y

, and if

X

has a

Γ

-universal set

U

parametrised by

Y

, then

X

has one parametrised by

Y^{'}

(2) Let $f : Y^{'} \to Y$ be a continuous surjection. If $X$ has a $Γ$ -universal set $U$ parametrised by $Y$ , then $X$ has a $Γ$ -universal set parametrised by $Y^{'}$ .

Proof. Ad (1): Let $U^{'}$ be a $Γ$ -set in $X \times Y^{'}$ such that $U^{'} \cap (X \times Y) = U$ . Then $U^{'}$ is a $Γ$ -universal set parametrised by $Y^{'}$ .

Ad (2): Let $U^{'} = {(x, y^{'}) : (x, f (y)) \in U} = {(i_{X} \times f)}^{- 1} (U)$ . Since $i_{X} \times f$ is continuous, $U^{'} \in Γ (X \times Y^{'})$ . If $A \in Γ (X)$ , then there is an element $y \in Y$ such that $A = U^{y}$ . Pick $η \in Y^{'}$ such that $y = f (η) \in Y$ . Then $A = U^{y} = U^{f (η)} = {(U^{'})}^{η}$ . $◊$

Lemma 4 Suppose

X

has a

Σ_{2}^{0}

-universal set,

U

, parametrised by

Y

. Let

f : X \to X^{'}

be a perfect map. Then

X^{'}

has a

Σ_{2}^{0}

-universal set,

U^{'}

, parametrised by

Y

Proof. The set $U^{'} = (f \times i_{Y}) (U)$ is a $Σ_{2}^{0}$ -universal set for $X^{'}$ parametrised by $Y$ . $◊$
The following result is easy to show.

Lemma 5 Let

Σ_{α}^{0} (X) = \cup_{i \in I} A_{i}

. Suppose that for each

i \in I

there is a

Σ_{α}^{0}

-set

U_{i}

X \times Y_{i}

such that each

A \in A_{i}

is equal to

U_{i}^{y}

for some

y \in Y_{i}

. If

| I | \leq ℵ_{0}

α \in ω

, then the set

\cup_{i \in I} U_{i}

is a

Σ_{α}^{0}

-universal set for

X

parametrised by

\oplus_{i \in I} Y_{i}

Borel universal sets exist: By [8, Theorem 22.3], if $X$ has countable weight, and $Y = 2^{ω}$ , then $X$ has $Γ$ -universal sets parametrised by $Y$ for any $Γ$ . The following result is in a similar spirit.

Proposition 6 Suppose

X

has an open universal set,

U

, parametrised by

Y

. Then for each

α \in ω_{1} - {0}

X

has a

Σ_{α}^{0}

-universal set,

U_{α}

, parametrised by

Y^{ω}

Proof. The first step is to show that $X$ has a $Π_{2}^{0}$ -universal set, $T$ , parametrised by $Y^{ω}$ . Define $T_{n} \subseteq X \times Y^{ω}$ by $T_{n} = {(x, f) : x \in U^{f (n)}} .$ Let $ψ_{n} : X \times Y^{ω} \to X \times Y$ be the continuous function that takes $(x, f)$ to $(x, f (n))$ . Then $ψ_{n}^{- 1} (U) = T_{n}$ . Since $ψ$ is continuous and $U$ is open, it must be the case that $T_{n}$ is also open. The set $T = \cap_{n \in ω} T_{n}$ is then a $Π_{2}^{0}$ -set of $X \times Y^{ω}$ , and is a $Π_{2}^{0}$ -universal set for $X$ parametrised by $Y^{ω}$ . The set $U_{2} = (X \times Y^{ω}) - T$ is then a $Σ_{2}^{0}$ -universal set for $X$ .

Suppose $α \in ω_{1} - {0}$ is a limit, and that for all $η \in α - {0}$ , we have defined $Σ_{η}^{0}$ -universal sets, $U_{η}$ , for $X$ parametrised by $Y^{ω}$ . Now $α$ is countable, and has a sequence ${η_{n}}_{n \in ω} \subseteq α such that for all n \in ω, η_{n} \leq η_{n + 1}, and sup {η_{n} + 1 : n \in ω} = α$ . Let $φ : N \times N \to N$ be a bijection, and $φ_{n} : N \to N$ be the function $φ_{n} (m) = φ (n, m)$ . We now define $T_{n} \subseteq X \times Y^{ω}$ by $T_{n} = {(x, f) : x \in U_{η_{n}}^{f \cdot φ_{n}}} .$ Let $ψ_{n} : X \times Y^{ω} \to X \times Y^{ω}$ be the continuous function that takes $(x, f)$ to $(x, f \cdot φ_{n})$ . Then $ψ_{n}^{- 1} (U_{η_{n}}) = T_{n}$ . As before, each $T_{n}$ is a $Σ_{η_{n}}^{0}$ -set in $X \times Y^{ω}$ . The set $T = \cap_{n \in ω} T_{n}$ is then a $Π_{α}^{0}$ -set, and a $Π_{α}^{0}$ -universal set for $X$ parametrised by $Y^{ω}$ .

If $α$ is a successor, we simply follow the arguments in the construction of $U_{2}$ given $U$ . $◊$

Corollary 7 Suppose

U

is an open set in

X \times Y

such that each open set of

X

is a countable union of the

U^{y}

’s. Then there is an open universal set for

X

parametrised by

Y^{ω}

Such a $U$ is called a $σ$ -generator for the open sets of $X$ parametrised by $Y$ . If the unions do not have to be countable, then we say that $U$ is a generator for the open sets of $X$ .

Lemma 8 Let

(X, τ)

be a space with weight

w (X)

. Then, for each

α \in ω_{1} - {0}

(1) $X$ has a $Σ_{α}^{0}$ -universal set parametrised by $D {(| τ |)}^{ω}$ ; and

(2) $X$ has a $Σ_{α}^{0}$ -universal set parametrised by $2^{w (X)}$ .

Proof. For any space $(X, τ)$ , $D (| τ |)$ clearly parametrises an open universal, and so, by Proposition 6, for any $α \in ω_{1}$ , $D {(| τ |)}^{ω}$ parametrises a $Σ_{α}^{0}$ -universal for $(X, τ)$ .

By Proposition 6 it suffices to show that $X$ has an open universal set parametrised by $2^{w (X)}$ . Let $λ = w (X)$ , and ${B_{α}}_{α \in λ}$ be a base for $X$ . Put $Y = 2^{λ}$ . Let $U = {(x, y) : x \in \cup {B_{α} : y (α) = 0}} .$ We only need to show that this is open in $X \times Y$ (as it is quite clear that all open sets of $X$ can be represented by a $U^{y}$ ). Let $(x, y) \in U$ . Then $x$ is in the open set $U^{y} = \cup {B_{α} : y (α) = 0}$ . Let $α_{0} \in λ$ be such that $y (α_{0}) = 0$ and $x \in B_{α_{0}}$ . The set $π_{α_{0}}^{- 1} ({0})$ is open in $Y$ . Let $(ξ, η) \in B_{α_{0}} \times π_{α_{0}}^{- 1} ({0})$ . Now, $η (α_{0}) = 0$ . So $U^{η} \supseteq B_{α_{0}}$ , and $(ξ, η) \in U$ . $◊$

Lemma 9 Suppose

X_{i}

has an open universal set

U_{i}

parametrised by

Y_{i}

, for each

i \in n

. Then the set

U = {({(x_{i})}_{i \in n}, {(y_{i})}_{i \in n}) : x_{i} \in U_{i}^{y_{i}}, y_{i} \in Y_{i}}

is a generator for the open sets of

\prod_{i \in n} X_{i}

parametrised by

\prod_{i \in i n} Y_{i}

Theorem 10 The following are equivalent for a cardinal

κ

(1) $2^{ℵ_{0}} < 2^{κ}$ ;

(2) If $X$ has a $Σ_{α}^{0}$ -universal set (for some $α \in ω_{1} - {0}$ ) parametrised by $Y$ , with $| Y | \leq c$ , then $h c (X) < κ$ ;

(3) Every space $X$ with an open universal set parametrised by a compact first countable separable space $Y$ has hereditary cellularity less than $κ$ .

2 Open universal sets

In this section we gather together some results mentioned in [5], relevant to the present discussion.

2.1 General spaces

Theorem 11 Suppose

X

has an open universal set,

U

, parametrised by

Y

. Then

(1) $w (X) \leq n w (Y)$ ;

(2) $h d (X) \leq h L (Y)$ ;

(3) $h L (X) \leq h d (Y)$ ;

(4) $h c (X) \leq h c (Y)$ .

Statements (2) and (3) hold because $U$ is an open universal set. We shall see later that if $U$ is a $Π_{2}^{0}$ -universal set, then both of them are consistently false.

Corollary 12 Suppose

X

has an open universal set parametrised by

Y

. Then for each

n \in ω

(1) $h d (X^{n}) \leq h L (Y^{n})$ ;

(2) $h L (X^{n}) \leq h d (Y^{n})$ ;

(3) $h c (X^{n}) \leq h c (Y^{n})$ .

Next we use Todorcevic’s spaces $A [\leq]$ , $A [\geq]$ and $A [\leq_{l e x}]$ to show that, in the countable case, the inequalities $h d (X) \leq h d (Y^{ω})$ , $h L (X) \leq h L (Y^{ω})$ and $n w (X) \leq max (h d (Y^{ω}), h L (Y^{ω}))$ , consistently need not hold, for a space $X$ with an open universal set parametrised by $Y$ . It is known that under the assumption that $S$ -spaces do not exist [15], it is consistent that if $Y$ is either hereditarily separable or hereditarily Lindelöf, then $X$ must be both hereditarily separable and hereditarily Lindelöf [5].

Example 13

(b = ℵ_{1})

Let

X = A [\geq_{l e x}]

and

Y^{'} = A [\leq_{l e x}] \times ω

. Then

X

has an open universal set parametrised by

Y = {(Y^{'})}^{ω}

, with

n w (X) > ℵ_{0}

and

Y

being hereditarily separable and hereditarily Lindelöf.

Example 14

(b = ℵ_{1})

(1) Let $X = A [\geq]$ and $Y^{'} = A [\leq] \times ω$ . Then $X$ has an open universal set parametrised by $Y = {(Y^{'})}^{ω}$ , with $X$ being an $L$ -space and $Y$ a strong $S$ -space.

(2) Let $X = A [\leq]$ and $Y^{'} = A [\geq] \times ω$ . Then $X$ has an open universal set parametrised by $Y = 2^{ω} \times {(Y^{'})}^{ω}$ , with $X$ being an $S$ -space and $Y$ a strong $L$ -space.

Example 15 Let

X = D^{I} [\geq]

and

Y^{'} = D^{I} [\leq] \times ω

. Then

X

has an open universal set parametrised by

Y = {(Y^{'})}^{ω}

, with

h d (X) > h L (X) = | D | = h d (Y^{'}) < h L (Y^{'}) .

2.2 Compact spaces

We note that given $X$ is compact, we obtain in addition to theorem 11, the inequalities $h d (X) \leq h d (Y)$ and $h L (X) \leq h L (Y)$ .

Lemma 16 Let

X

be a compact set that has a closed universal set,

U

, parametrised by

Y

. Then

(1) the set-valued map $Φ : y \to U^{y}$ defined on $Y$ is upper semi-continuous; and

(2) if $Y^{(1)} = {y \in Y : | Φ (y) | = 1}$ and $φ : Y^{(1)} \to X$ be the map that picks out the unique element of $Φ (y)$ , then $φ$ is a continuous map.

Corollary 17 Let

X

be a compact space that has an open universal set,

U

, parametrised by

Y

. Let

P

be any hereditary property preserved by taking continuous images. Then, if

Y^{n}

has

P

, so does

X^{n}

Hence, in the situation above $h d (X^{n}) \leq h d (Y^{n})$ and $h L (X^{n}) \leq h L (X^{n})$ . This result, but not the existence of a continuous map, will be extended from open universal sets to $Σ_{n}^{0}$ -universal sets $(0 < n < ω)$ in Theorem 24.

3 Universal sets of finite Borel classes

Let $X$ be a space. We now consider when such a space has a $Σ_{n}^{0}$ -universal set parametrised by a second countable $Y$ , for $1 < n < ω$ .

3.1 General spaces

Theorem 18 For a topological space

X

and

n \in ω - {0}

X

has a

Σ_{n}^{0}

-universal set parametrised by a space

Y

of weight

κ

if and only if it has a coarser

T_{1}

-topology of weight no greater than

κ

with the same

Σ_{n}^{0}

-sets. In particular, if

w (Y) = ℵ_{0}

, then

ψ (X) = ℵ_{0}

Proof. Suppose $(X, τ)$ has a coarser $T_{1}$ -topology $υ$ of weight $w (υ)$ such that every $τ$ - $Σ_{n}^{0}$ -set is an $υ$ - $Σ_{n}^{0}$ -set. By Lemma 8 and Proposition 6, $(X, υ)$ has a $Σ_{n}^{0}$ -universal set parametrised by $2^{w (υ)}$ . As $υ \subseteq τ$ and $Σ_{n}^{0} (τ) = Σ_{n}^{0} (υ)$ , we conclude that the same set is a $Σ_{n}^{0}$ -universal set for $(X, τ)$ parametrised by $2^{w (υ)}$ , which has weight $w (υ)$ .

If $n$ is odd and greater than 1, then we consider $U$ as a $Σ_{n}^{0}$ -universal set for $X$ parametrised by $Y$ , which has base $B$ of size $κ$ . The case $n = 1$ has already be proved in Theorem 11(1). There exists a countable collection of open sets, ${U_{f} : f \in ω^{n}}$ , of the space $X \times Y$ such that

\begin{matrix} \begin{matrix} U = \cup_{m_{0} \in ω} U_{(m_{0})}, \\ U_{(m_{0})} = \cap_{m_{1} \in ω} U_{(m_{0}, m_{1})}, for m_{0} \in ω, \\ U_{(m_{0}, m_{1})} = \cup_{m_{2} \in ω} U_{(m_{0}, m_{1}, m_{2})}, for m_{0}, m_{1} \in ω, \\ . \\ U_{(m_{0}, \dots, m_{n - 2})} = \cap_{m_{n - 1} \in ω} U_{(m_{0}, \dots, m_{n - 2}, m_{n - 1})}, for m_{0}, \dots, m_{n - 2} \in ω . \end{matrix} \end{matrix}

(0) Define for each $B \in B$ and $f \in ω^{n}$ the open sets, $V_{f} (B) = π_{X} (\cup {V \times B : V \times B \subseteq U_{f} and V is open in X})$ and $C$ the collection (of size $κ$ ) of all the finite intersections of them. Then $C$ is a $T_{1}$ -point separating family of $τ$ -open sets. Let $x_{1} \neq x_{2 </}$