Proceedings of the Ninth Prague Topological Symposium Contributed papers from the symposium held in Prague, Czech Republic, August 19–25, 2001

QUASIORDERS ON TOPOLOGICAL CATEGORIES

VĚRA TRNKOVÁ

Abstract. We prove that, for every cardinal number α ≥ 𝔠, there exists a metrizable space X with |X| = α such that for every pair of quasiorders ≤_{1}, ≤_{2} on a set Q with |Q| ≤ α satisfying the implication

q ≤_{1} q′ ⟹ q ≤_{2} q′

there exists a system {X(q) : q ∈ Q} of non-homeomorphic clopen subsets of X with the following properties:

• q ≤_{1} q′ if and only if X(q) is homeomorphic to a clopen subset of X(q′),

• q ≤_{2} q′ implies that X(q) is homeomorphic to a closed subset of X(q′) and

• ¬(q ≤_{2} q′) implies that there is no one-to-one continuous map of X(q) into X(q′).

1. Introduction and the Main Result

Let M be a class of morphisms of a category K containing all isomorphisms and closed with respect to the composition. Then M determines a quasiorder ≼ on the class of objects of K by the rule

X ≼ Y if and only if there exists m ∈ M with the domain X and the codomain Y.

An M-representation of a quasiordered set (Q, ≤) in K is any collection X = {X(q) : q ∈ Q} of non-isomorphic objects of K such that, for every q, q′ ∈ Q,

q ≤ q′ if and only if X(q) ≼ X(q′).

2000 Mathematics Subject Classification. 54B30, 54H10. Key words and phrases. homeomorphism onto clopen subspace, onto closed subspace,

quasiorder, metrizable spaces.

Financial support of the Grant Agency of the Czech Republic under the grants no.201/99/0310 and 201/00/1466 is gratefully acknowledged. Supported also by MSM 113200007.

Věra Trnková, Quasiorders on topological categories, Proceedings of the Ninth Prague Topological Symposium, (Prague, 2001), pp. 321–330, Topology Atlas, Toronto, 2002;

arXiv:math.GN/0204143.

Which quasiordered sets have M-representations in which categories K for which classes M? In topology, this investigation with M being the class of all homeomorphic embeddings has rather long tradition. In 1926, C. Kuratowski and W. Sierpiński proved (see [2, 3]) that the antichain of the cardinality 2^{𝔠} and the ordinal 𝔠^{+} have such representations within the category of all subspaces of the real line ℝ. In 1993, P.T. Matthews and T.B.M. McMaster refreshed this field of problems and proved (see [5]) that every partially ordered set of the cardinality 𝔠 has such representation. In 1999, A.E. McCluskey, T.B.M. McMaster and W.S. Watson proved (see [8]) that the set (exp ℝ, ⊆) of all subsets of the real line ℝ ordered by the inclusion also has such a representation, i.e. a representation by subspaces of ℝ with respect to the embeddability. Since (exp ℝ, ⊆) contains an antichain of the cardinality 2^{𝔠}, their result implies the first result of C. Kuratowski and W. Sierpiński. Analogously it implies the previous result of P.T. Matthews and T.B.M. McMaster because (exp ℝ, ⊆) contains an isomorphic copy of any partially ordered set of cardinality at most 𝔠. The authors of [8] also announced that they have a counterexample consistent with ZFC to the statement that every partially ordered set of the cardinality 2^{𝔠} can be represented by subspaces of ℝ with respect to the embeddability. In [6], A.E. McCluskey and T.B.M. McMaster generalized the construction of [8] and they proved that, for any infinite cardinal numbers 𝔞, 𝔟 such that 𝔟^{𝔞} = 𝔟, every partially ordered set (P, ≤) with |P | ≤ 𝔟 has a representation (with respect to the embeddability) by subsets of every T_{3}-space X containing a dense subset D with |D| ≤ 𝔞 whenever every non-empty open set of X has the cardinality 𝔟. Moreover, if X admits a homeomorphism into itself such that every x ∈ X has an infinite orbit, then every quasiordered set obtained from a partially ordered set (P, ≤) with |P | ≤ 𝔟 by splitting every p ∈ P into 2^{𝔟} mutually comparable points has also such a representation.

Given a cardinal number α, A.E. McCluskey and T.B.M. McMaster also present (in [7]) a construction of a T_{0}-space X with |X| = δ(α) such that every quasiordered set (Q, ≤) with |Q| ≤ α has a representation by subsets of X (with respect to the embeddability), where δ(α) is the smallest cardinal number δ such that there exist α distinct cardinal numbers γ (not necessarily infinite) smaller than δ. If α = ℵ_{0}, then δ(α) = ℵ_{0}; hence there exists a countable T_{0}-space X such that every countable quasiordered set has a representation (with respect to the embeddability) by subsets of X. For α uncountable, the size of X is rather high. The statement below offers a stronger result for any α ≥ 𝔠.

For every cardinal number α ≥ 𝔠 there exists a metrizable space X with |X| = α such that every quasiordered set (Q, ≤) with |Q| ≤ α has an Mrepresentation by retracts of X whenever

(1) M consists of one-to-one continuous maps and it contains all coproduct injections (= homeomorphic embeddings onto clopen subspaces) or (2) M consists of continuous surjections and it contains all product pro-

jections. The claim (1) implies e.g. the representability of every quasiordered set

(Q, ≤) with |Q| ≤ α by retracts of X with respect to the embeddability or the embeddability onto closed subspaces or onto retracts or onto clopen subspaces. The claim (2) implies the representability e.g. by being continuous image or a quotient or a continuous open image or a factor in a product. Here we prove a result much stronger than (1) above, namely the following.

Theorem. For every cardinal number α ≥ 𝔠, there exists a metrizable space X with |X| = α such that, for every pair ≤_{1}, ≤_{2} of quasiorders on a set Q with |Q| ≤ α satisfying the implication

q ≤_{1} q′ ⟹ q ≤_{2} q′,

there exists a system {X(q) : q ∈ Q} of non-homeomorphic clopen subsets of X with the following properties:

(1) q ≤_{1} q′ if and only if X(q) is homeomorphic to a clopen subset of X(q′),

(2) q ≤_{2} q′ implies that X(q) is homeomorphic to a closed subset of X(q′) and

(3) ¬(q ≤_{2} q′) implies that there is no one-to-one continuous map of X(q) into X(q′).

The proof of this theorem is presented in the part II of the present paper. Inspecting the proof, one can see that none of the spaces X(q) has an isolated point. Hence if one adds α isolated points to every X(q) (and also to the space X), she (he) gets the following result:

For every cardinal number α ≥ 𝔠 there exists a metrizable space X with |X| = α such that every quasiordered set (Q, ≤) with |Q| ≤ α has an Mrepresentation by clopen subspaces of X whenever M is the class of all continuous bijections.

We do not present here the proof of the above claim (2) and of its stronger variant concerning simultaneous representation of a pair ≤_{1}, ≤_{2} of quasiorders by product projections and continuous surjections. The proof will appear elsewhere.

Finally, let us refresh here some results which have applications in the present field of problems. In [1], J. Adámek and V. Koubek introduced a sum-productive representation of a partially ordered commutative semigroups as follows: If (S, ∘, ≤) is a partially ordered commutative semigroup (i.e. if (S, ∘) is a commutative semigroup and ≤ is a partial order on S such that

(a ≤ b) and (a′ ≤ b′) ⟹ a ∘ a′ ≤ b ∘ b′),

then its sum-productive representation in a category K is any collection {X(s) : s ∈ S} of objects of K such that

(×) X(s ∘ s′) is always isomorphic to the product X(s) × X(s′) and (≤) s ≤ s′ if and only if X(s) is isomorphic to a summand of X(s′) (i.e.

X(s′) is a coproduct of X(s) and an object of K). In [11], J. Vinárek proved that every partially ordered commutative semi-

group has a sum-productive representation in the category of all metric zero-dimensional spaces. Every partially ordered set (P, ≤) can be enlarged to a partially ordered set (S, ≤) in which every pair of elements has an infimum. Putting p∘p′ = inf{p, p′}, one gets the partially ordered commutative semigroup (S, ∘, ≤) and the Vinárek’s result can be applied. Hence

every partially ordered set has a representation by zero-dimensional metrizable spaces with respect to the embeddability onto clopen subspaces.

In [10], sum-productive representations in the category of all F_{σδ}-and-G_{σδ} subspaces of the Cantor discontinuum are examined. Omitting the product forming again, we get the result that

if ℂ is a countable set, then (exp ℂ, ⊆) has a representation by F_{σδ} and G_{σδ} subspaces of the Cantor discontinuum with respect to the embeddability onto clopen subspaces.

These scattered results are surrounded by many unsolved questions, such as: which quasiordered sets can be represented by metrizable zero-dimen-sional spaces or by closed or Borelian or all subsets of the real line with respect to the embeddability onto clopen subspaces or onto Borelian subspaces or onto closed subspaces or onto retracts; and many others.

2. Proof of the Theorem

Let α ≥ 𝔠 be given. Let Q be a set with |Q| = α. In the part A of the proof, we suppose that we have sets S^{(1)}, …, S^{(4)} of metrizable spaces of cardinality α with the five properties below and we prove Theorem using such sets. In the part B, we prove that such sets S^{(1)}, …, S^{(4)} really do exist.

Part A.

a. Thus, let us suppose that S^{(1)}, …, S^{(4)} are sets of metrizable spaces with |S^{(i)}| = α and |Y | = α for every Y ∈ S^{(i)}, i = 1, …, 4, such that the statements (1)–(5) below are satisfied:

(1) S^{(1)} = {A_{q}, B_{q} : q ∈ Q} are spaces such that, for every q ∈ Q, A_{q} is homeomorphic to a clopen subspace of B_{q} and B_{q} is homeomorphic to a closed subspace but to no clopen subspace of A_{q}; moreover, if q ≠ q′, there exists no continuous one-to-one map of any A_{q}, B_{q} into any A_{q′}, B_{q′};

(2) S^{(2)} = {D_{q}, E_{q} : q ∈ Q} are such that D_{q} is homeomorphic to a closed subspace of E_{q} and E_{q} is homeomorphic to a closed subspace of D_{q} but there exists no homeomorphism of D_{q} onto a clopen subspace of E_{q} and no homeomorphism of E_{q} onto a clopen subspace of D_{q}; moreover, if q ≠ q′, then there exists no continuous one-to-one map of any D_{q}, E_{q} into any D_{q′}, E_{q′}; (3) S^{(3)} = {M_{q}, N_{q} : q ∈ Q} are spaces such that M_{q} is homeomorphic

to a clopen subspace of N_{q} and N_{q} is homeomorphic to a closed subspace but to no clopen subspace of M_{q} (i.e. they are mutually situated as A_{q} and B_{q} in (1)); moreover, if q ≠ q′, there exists no continuous one-to-one map of any M_{q}, N_{q} into any M_{q′}, N_{q′};

(4) S^{(4)} = {G_{q}, H_{q} : q ∈ Q} are spaces such that G_{q} is homeomorphic to a clopen subspace of H_{q} and H_{q} is homeomorphic to a clopen subspace of G_{q} but G_{q} is not homeomorphic to H_{q}; moreover, if q ≠ q′, there exists no continuous one-to-one map of any G_{q}, H_{q} into any G_{q′}, H_{q′};

(5) if i, j = 1, …, 4, i ≠ j, Z ∈ S^{(i)}, Y ∈ S^{(j)}, then there exists no continuous one-to-one map of Z into Y.

We put

X = ∐_{q∈Q} (A_{q} ∐ B_{q} ∐(D_{q} × ω) ∐(E_{q} × ω) ∐ M_{q} ∐ N_{q} ∐ G_{q})

where ∐ and ∐ denote the coproduct (= disjoint union as clopen subspaces) and ω is a countable discrete space (i.e. D_{q} × ω is a coproduct of countably many copies of D_{q} and analogously for E_{q}).

Hence X is a metrizable space and |X| = α. We show that X has all the required properties. In the reasoning below, we shall frequently use the statement (5) without mentioning it.

b. Let (Q, ≤_{1}, ≤_{2}) be a set with two quasiorders ≤_{1}, ≤_{2} such that

q ≤_{1} q′ ⟹ q ≤_{2} q′.

To begin with, let us suppose, moreover, that ≤_{1} is a partial order, i.e.

(2.1)  (q ≤_{1} q′) and (q′ ≤_{1} q) ⟹ q = q′

(this requirement will be removed at the end of part A of the proof).

By means of S^{(1)}–S^{(3)}, we construct a system {X(q) : q ∈ Q} of clopen subspaces of X such that

• q ≤_{1} q′ if and only if X(q) is homeomorphic to a clopen subset of X(q′),

• q ≤_{2} q′ if and only if X(q) is homeomorphic to a closed subset of X(q′) and

• if ¬(q ≤_{2} q′), then there is no continuous one-to-one map of X(q)

into X(q′). First, let us define < by the rule

q < q′ if and only if (q ≤_{1} q′) and (¬(q′ ≤_{1} q)) and (q′ ≤_{2} q),

and q ≤ q′ denotes q < q′ or q = q′. (By (2.1), ≤ is a partial order.) Let C be the system of all components of ≤ (i.e. q, q′ ∈ C ∈ C if and only if there exist q_{0}, …, q_{n} in Q such that q = q_{0} ≤ q_{1} ≥ q_{2} ≤ … q_{n} = q′).

For every q ∈ C ∈ C put

\begin{matrix}S_{q}^{(1)} = \{B_{q′} : q′ ≤ q\} ∪ \{A_{q′} : q′ ∈ C and ¬(q′ ≤ q)\}, \\ X_{q}^{(1)} = ∐ S_{q}^{(1)}. \end{matrix}

Moreover, let us denote

B_{C} = {B_{q} : q ∈ C},
D_{C} = {D_{q} × ω : q ∈ C},
E_{C} = {E_{q} × ω : q ∈ C}.

Observation. If q ∈ C and q′ ∈ C′ with C, C′ ∈ C, C ≠ C′, then there exists no continuous one-to-one map of X_{q}^{(1)} into X_{q′}^{(1)}. If q, q′ ∈ C, then

q < q′ if and only if X_{q}^{(1)} is homeomorphic to a clopen subspace of X_{q′}^{(1)} and

X_{q′}^{(1)} is homeomorphic to a closed but not to any clopen subspace of X_{q}^{(1)}.

c. On Q, let us define q R q′ if and only if either

q < q′ or
q ≤_{2} q′ and q′ ≤_{2} q but neither q ≤_{1} q′ nor q′ ≤_{1} q.

Let ≺ be the transitive envelope of R, and let ⪯ mean ≺ or =. Then ⪯ is a quasiorder on Q; let L be the system of all its components. Clearly, each C ∈ C is a subset of precisely one L ∈ L. For every q ∈ C ∈ C, C ⊆ L, put

\begin{matrix}S_{q}^{(2)} = S_{q}^{(1)} ∪ ⋃_{C′∈C,C≠C′⊆L}(B_{C′} ∪ E_{C′} ∪ D_{C′}) ∪ E_{C}; \\ X_{q}^{(2)} = ∐ S_{q}^{(2)} \end{matrix}

Observation. If q, q′ ∈ C ∈ C, q ≠ q′, then S_{q}^{(2)} differs from S_{q}^{(1)} by the same summand

⋃_{C′∈C,C≠C′⊆L}(B_{C′} ∪ E_{C′} ∪ D_{C′}) ∪ E_{C}

as S_{q′}^{(2)} differs from S_{q′}^{(1)}. Hence X_{q}^{(2)} and X_{q′}^{(2)} are in the same relations

as X_{q}^{(1)} and X_{q′}^{(1)}. If C, C′ ∈ C, C ≠ C′ and both C, C′ are subsets of L ∈ L, q ∈ C, q′ ∈ C′, then X_{q} is homeomorphic to a closed subspace but not to any clopen subspace of X_{q′} and vice versa. This is because ∐(E_{C} ∪ D_{C} ∪ E_{C′}) is homeomorphic to a closed (but not to any clopen!) subspace of ∐(E_{C′} ∪D_{C′} ∪E_{C}), by (2) (and ∐(S_{q}^{(1)} ∪B_{C′}) is homeomorphic to a closed subspace of ∐(S_{q′}^{(1)} ∪ B_{C}), by (1)).

Hence {X_{q}^{(2)} : q ∈ L} is such that

q ≤_{1} q′ if and only if X_{q}^{(2)} is homeomorphic to a clopen subset of X_{q′}^{(2)}

and

q ≤_{2} q′ if and only if X_{q}^{(2)} is homeomorphic to a closed subset of X_{q′}^{(2)}.

(For every q, q′ ∈ L, we have q ≤_{2} q′ and q′ ≤_{2} q, of course.) Moreover, if L, L′ ∈ L, L ≠ L′, q ∈ L, q′ ∈ L′, then there exists no one-to-one continuous map of X(q) into X(q′) and vice versa. Let us denote

S_{L}^{(2)} = {B_{C} ∪ E_{C} ∪ D_{C} : C ∈ C, C ⊆ L}.

d. On Q, let us define qṘq′ if and only if either

q ≺ q′ or
q ≠ q′, q ≤_{1} q′ or
q ≠ q′, q ≤_{2} q′.

Let ˙≺ be the transitive envelope of Ṙ, and let ˙⪯ mean ˙≺ or =. Let T be the system of all components of the quasiorder ˙⪯. Every L ∈ L is a subset of precisely one T ∈ T . If L, L′ ∈ L, L ≠ L′, q ∈ L, q′ ∈ L′ and q ≤_{2} q′, then for no q̄ ∈ L, q̄′ ∈ L′ is q̄′ ≤_{2} q̄ because L ≠ L′. Hence in fact, ˙⪯ determines a partial order on the components in L, let us denote it also by ˙⪯. For every L ∈ L define

S̃_{q}^{(2)} = ⋃_{L′ ˙≺L} S̃_{L′}^{(2)}

and for q ∈ L,

\begin{matrix}S̃_{q}^{(2)} = S_{q}^{(2)} ∪ S̃_{L}^{(2)}, \\ X̃_{q}^{(2)} = ∐ S̃_{q}^{(2)}. \end{matrix}

If q ∈ L, q′ ∈ L′ and L, L′ are components of L incomparable in the partial order ˙⪯, then there is no continuous one-to-one map of X̃_{q}^{(2)} into X̃_{q′}^{(2)} and

vice versa. If q, q′ ∈ L, then X̃_{q}^{(2)} is in the same position with respect to X̃_{q′}^{(2)} as X_{q}^{(2)} to X_{q′}^{(2)}. If q′ ∈ L′ ˙⪯L ∋ q, then X̃_{q′}^{(2)} is homeomorphic to

a clopen subspace of X̃_{q}^{(2)} and there exists no continuous one-to-one map of X̃_{q}^{(2)} into X̃_{q′}^{(2)}. This is what we need whenever q′ ≤_{1} q. However, if q′ ≤_{2} q

and ¬(q′ ≤_{1} q), we still have to modify the systems S̃_{q}^{(2)} and the spaces X̃_{q}^{(2)} by means of the spaces in S^{(3)} in (3). For q ∈ L, we put

\begin{matrix}S_{q}^{(3)} = S̃_{q}^{(2)} ∪ \{N_{q′} : q′ ∈ T and q′ ≤_{1} q\} ∪ \{M_{q′} : q′ ∈ T and ¬(q′ ≤_{1} q)\}; \\ X(q) = ∐ S_{q}^{(3)}. \end{matrix}

First, we outline why these “new summands” do not destroy the mutual position of X̃_{q_{1}}^{(2)} and X̃_{q_{2}}^{(2)} whenever q_{1}, q_{2} ∈ L ⊆ T:

• q_{1} ≤_{1} q_{2}: if N_{q′} ∈ S_{q_{1}}^{(3)}, i.e. q′ ≤_{1} q_{1}, we get q′ ≤_{1} q_{2} so that N_{q′} ∈ S_{q_{2}}^{(3)}; since S_{q_{2}}^{(3)} always contains either M_{q′} or N_{q′}, the case M_{q′} ∈ S_{q_{1}}^{(3)} is clear.

• q_{1} ≤_{2} q_{2}: if N_{q′} ∈ S_{q_{1}}^{(3)}, i.e. q′ ≤_{1} q_{1}, then either q′ ≤_{1} q_{2} hence N_{q′} ∈ S_{q_{2}}^{(3)}, or ¬(q′ ≤_{1} q_{2}) hence M_{q′} ∈ S_{q_{2}}^{(3)}; but N_{q′} is homeomorphic to a closed subspace of M_{q′}.

No other case for q_{1}, q_{2} ∈ L is possible (if ¬(q_{1} ≤ q_{2}) for q_{1}, q_{2} ∈ L, then already S̃_{q_{1}}^{(2)} is not homeomorphic to a clopen subspace of S̃_{q_{2}}^{(2)}).

Next we show that the system {X(q) : q ∈ Q} has all the required properties.

Let q_{1}, q_{2} ∈ Q. If q_{1}, q_{2} ∈ L, we have just proved it. If q_{1} ∈ T, q_{2} ∈ T′ where T, T′ are distinct elements of T (i.e. distinct components of the quasiorder ˙⪯), then there is no continuous one-to-one map of X(q_{1}) into X(q_{2}) or vice versa. But this precisely corresponds to the fact that ¬(q_{1} ≤_{2} q_{2}) and ¬(q_{2} ≤_{2} q_{1}). The remaining case is that q_{1}, q_{2} are in distinct components L_{1}, L_{2} of L but both L_{1} and L_{2} are subsets of a component T ∈ T . We discuss the following cases:

α) L_{1} and L_{2} are incomparable in the partial order ˙⪯: then there exists no one-to-one continuous map of X̃_{q_{1}}^{(2)} into X̃_{q_{2}}^{(2)} and adding any summands M_{q′}, N_{q′} cannot change this situation; hence there exists no continuous one-to-one map of X(q_{1}) into X(q_{2}) or vice versa. β) let L_{1} ˙≺L_{2}: hence X̃_{q_{1}}^{(2)} is homeomorphic to a clopen subspace of X̃_{q_{2}}^{(2)} and there exists no one-to-one continuous map of X̃_{q_{2}}^{(2)} into X̃_{q_{1}}^{(2)}, adding of any summands M_{q′}, N_{q′} cannot change the latter fact; since ¬(q_{2} ≤_{2} q_{1}), this is precisely what we need. Thus we have to discuss the cases q_{1} ≤_{1} q_{2} and q_{1} ≤_{2} q_{2}. However the reasoning is precisely the same as in the above discussed cases when q_{1}, q_{2} ∈ L and q_{1} ≤_{1} q_{2} or q_{1} ≤_{2} q_{2}.

e. Finally, we have to remove the condition (2.1): If ≤_{1} is a quasiorder, we use the standard trick that we define an equivalence on Q by the rule

q ∼ q′ if and only if q ≤_{1} q′ and q′ ≤_{1} q.

Then ≤_{1} determines a partial order and ≤_{2} a quasiorder on the set Q/∼; let us denote them by ≤_{1} and ≤_{2} as well. For every [q] ∈ Q/∼, we construct S_{[q]}^{(3)} and X_{[q]}^{(3)} = ∐ S_{[q]}^{(3)} as described and, for q ∈ [q], we put

S_{q}^{(3)} = S_{[q]}^{(3)} ∪ {G_{q′} : q′ ∈ Q \ {q}} ∪ {H_{q}}

where G_{q′}, H_{q} are from S^{(4)}, i.e. they satisfy the condition (4). Then, for X_{q} = ∐ S_{q}^{(3)}, the system {X_{q} : q ∈ Q} has all the required properties.

Part B.

To finish the proof, it remains to show that the systems S^{(1)}, …, S^{(4)} with the properties (1)–(5) do exist. For this, we use the fact (see [9]) that for every α ≥ 𝔠 there exists a system 𝕊 of metrizable spaces of the cardinality α such that for every Y, Z ∈ 𝕊 and every continuous map f : Y → Z, either f is constant or Y = Z and f is the identity, and the system 𝕊 itself is large enough: for our purpose it suffices |𝕊| = α (although, as shown in [9], such 𝕊 with |𝕊| = 2^{α} does exist). Let

{A_{q}^{∗}, D_{q}^{∗}, E_{q}^{∗}, M_{q}^{∗}, G_{q}^{∗} : q ∈ Q}

be a subsytem of 𝕊.

(1) Choose two distinct points, say a_{q}, b_{q}, in the space A_{q}^{∗} and, in the space A_{q}^{∗} × ω, identify (b_{q}, n) with (a_{q}, n + 1). The obtained space is A_{q}, then B_{q} = A_{q}^{∗} ∐ A_{q}. Clearly, S^{(1)} = {A_{q}, B_{q} : q ∈ Q} satisfies (1).

(2) Choose two distinct points, say d_{q}^{(1)}, d_{q}^{(2)} in D_{q}^{∗} and e_{q}^{(1)}, e_{q}^{(2)} in E_{q}^{∗}; in D_{q}^{∗} × ω, identify (d_{q}^{(2)}, n) with (d_{q}^{(1)}, n + 1) and denote D̃_{q} the obtained space; analogously define Ẽ_{q}; then put

\begin{matrix}D_{q} = D_{q}^{∗} ∐ D̃_{q} ∐ Ẽ_{q} \\ E_{q} = E_{q}^{∗} ∐ D̃_{q} ∐ Ẽ_{q} \end{matrix}

Then S^{(2)} = {D_{q}, E_{q} : q ∈ Q} satisfies (2).

(3) The spaces M_{q} and N_{q} are constructed from M_{q}^{∗} as A_{q} and B_{q} from A_{q}^{∗} in (1). Then S^{(3)} = {M_{q}, N_{q} : q ∈ Q} satisfies (3).

(4) Let K be a closed subset of the Cantor discontinuum such that K is homeomorphic to K ∐ K ∐ K but not to K ∐ K. Such a space was constructed in [4]. Put G_{q} = K × G_{q}^{∗} and H_{q} = (K ∐ K) × G_{q}^{∗}. Since K is zero-dimensional while G_{q}^{∗} is connected and every continuous map of G_{q}^{∗} into itself is either the identity or a constant, every continuous one-to-one map f : G_{q} → H_{q} sends every K × {a} into (K ∐ K) × {a} for an arbitrary a ∈ G_{q}^{∗}. Since K and K ∐ K are non-homeomorphic, the space G_{q} is not homeomorphic to H_{q}. However G_{q} is homeomorphic to a clopen subspace of H_{q} and vice versa.

(5) The system S^{(1)} ∪ S^{(2)} ∪ S^{(3)} ∪ S^{(4)} satisfies (5) because

{A_{q}^{∗}, D_{q}^{∗}, E_{q}^{∗}, M_{q}^{∗}, N_{q}^{∗}, G_{q}^{∗} : q ∈ Q} ⊆ 𝕊,

evidently.

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