Subspaces of pseudoradial spaces

Martin Sleziak

October 25, 2005

Abstract We prove that every topological space (T_{0}-space, T_{1}-space) can be embedded in a pseudoradial space (in a pseudoradial T_{0}-space, T_{1}-space). This answers the Problem 3 in [2]. We describe the smallest coreflective subcategory A of Top such that the hereditary coreflective hull of A is the whole category Top.

1 Preliminaries

I’d like to present some results of my paper Subspaces of pseudoradial spaces [13]. I will start with defining some basic notions we will need later.

Transfinite sequence (x_{ξ})_{ξ<α} is a net defined on an infinite ordinal. We could say that it is a sequence indexed by ordinals less than α and the convergence (limit) of transfinite sequences in topological spaces is defined similarly as the usual convergence of sequences: A transfinite sequence (x_{ξ})_{ξ<α} converges to the point x ∈ X provided that:

∀_{U∈T} x ∈ U ∃_{η} η < α ∀_{ξ} η < ξ < α x_{ξ} ∈ U.

A topological space X is said to be pseudoradial if a subset V ⊆ X is closed in X whenever V is closed with respect to limits of transfinite sequences of elements of V. Pseudoradial spaces are also called chain-net spaces. We denote by Ã the smallest set such that A ⊆ Ã and Ã is closed with respect to limits of transfinite sequences. We will call the set Ã the pseudoradial closure of A. Obviously, a topological space X is pseudoradial if and only if for each subset A ⊂ X the equality \overline{A} = Ã holds.

An important example of a pseudoradial space is the space C(α), where α is a regular cardinal. C(α) is the topological space on the set α ∪ {α} such that a subset U ⊆ α ∪ {α} is open in C(α) if and only if U ⊆ α or card C(α) \ U < α. Equivalently a base for the topology of C(α) is B = {B_{ξ}; ξ < α} with B_{ξ} = {η ∈ α ∪ {α} : η ≥ ξ}.

Next we recall the notion of coreflective subcategories of Top. We will assume that all subcategories are full and isomorphism-closed, so a subcategory A of Top corresponds to a subclass of the class of all topological spaces (A is determined by the class of its objects). Then a coreflective subcategory of Top is a class of topological spaces which is closed under the formation of topological sums and quotient spaces. It can be seen directly from this definition that any intersection of coreflective subcategories of Top is again a coreflective subcategory of Top. Hence for each class A of topological spaces there exists the smallest coreflective subcategory of Top containing A. It is called the coreflective hull of A and denoted by CH(A). CH(A) contains precisely the quotient spaces of topological sums of spaces from A.

If a coreflective subcategory of Top is closed also under the formation of subspaces, it is called hereditary. Let A be a subcategory of Top and let SA denote the subcategory of Top consisting of all subspaces of spaces from A. It is known that if A is a coreflective subcategory of Top, then SA is a coreflective subcategory of Top as well.

We denote by PsRad the subcategory of all pseudoradial spaces. It is known that this subcategory is coreflective in Top and PsRad = CH({C(α); α is a regular cardinal}). Hence the subcategory SPsRad of all subspaces of pseudoradial spaces is coreflective as well.

We say that a topological space is a prime space, if it has only one accumulation point. The category Top is coreflective hull of prime spaces. (Every topological space X can be obtained as a quotient space of its prime factors X_{a}. X_{a} is the space constructed by making each point, other then a, isolated with a retaining its original neighborhoods. I.e. a subset U ⊆ X is open in X_{a} if and only if a ∉ U or there exists an open subset V of X such that a ∈ V ⊆ U.)

2 Overview of results

In the paper [2] the authors asked whether every topological space is a subspace of a pseudoradial space. We show that this is true. This result can be formulated using coreflective subcategories of Top like this: SPsRad = Top.

It suffices to show that all prime spaces belong to SPsRad, since Top is the coreflective hull of prime spaces.

This question was partly answered by Jin-Yuan Zhou in [14], where he proved under an additional set-theoretical assumption that the answer is affirmative for countable tight spaces. We show the same in ZFC in a similar way as in that paper. We first prove that every prime space can be embedded in a pseudoradial space and this implies (as we have already mentioned) that Top = SPsRad.

Zhou proved in his paper also something more. He showed (under an additional set-theoretical assumption) that every countable Hausdorff prime space can be embedded in a Hausdorff pseudoradial space. I don’t know whether this result holds in ZFC. It is proved that it holds for T_{1}-spaces in ZFC. It is known that embedding of prime spaces of the form ω ∪ {p}, p ∈ βω \ ω, into regular spaces is ZFC-independent (see [4]).

According to the results of the first part of this paper PsRad is an example of coreflective subcategory of Top such that SA = Top. We find the smallest subcategory A_{0} of Top having this property. This can be regarded as a kind of characterization of such subcategories. This is connected with the Problem 7 from [9]: To investigate coreflective subcategories of Top such that SA = Top and the hereditary coreflective kernel of A is the subcategory FG (i.e. A doesn’t contain any hereditary coreflective subcategory larger then FG).

3 Subspaces of pseudoradial spaces

Let us note that results of this section has been previously observed in [10].

We denote by S the Sierpiński space, i.e. the space defined on the set {0, 1} with the topology consisting of the empty set, the set {0} and the whole space. The key step of proving that each topological space is a subspace of a pseudoradial space is this proposition:

Proposition 3.2. If β is an infinite cardinal, then the topological power S^{β} of the space S is a pseudoradial space.

Proof. We should show that if a point t of S^{β} belongs to \overline{U} for some subset of U then it also belongs to Ũ. We show it only for t_{0} ∈ S^{β} where t_{0}(η) = 0 for each η ∈ β. The proof for other points of S^{β} if fairly similar.

So we assume that t_{0} ∈ \overline{U} and t_{0} ∉ U. Let us put for any subset A ⊆ β

f_{A}(x) = \begin{cases}0, & x ∈ A, \\ 1, & x ∉ A. \end{cases}

(I.e. f_{A} is the characteristic function of the set β \ A.) Each f_{A} is a point of S^{β} and t_{0} = f_{β} in this notation. Using transfinite induction we show that f_{A} ∈ Ũ for each A ⊆ β.

First, let A be a finite subset of β. Then U_{A} = {g ∈ S^{β}; g(x) = 0 for each x ∈ A} is a neighborhood of t_{0}, hence there exists a point g ∈ U_{A} ∩ U. The constant sequence (g)_{ξ<ω} converges to f_{A}.

Let card A = α ≤ β and for each B with card B < α it holds f_{B} ∈ Ũ. The set A can be written as an union of an increasing system A = ⋃_{ξ<α} B_{ξ} such that card B_{ξ} < α. (We can obtain such system using any well-ordering of A.) Obviously the transfinite sequence (f_{B_{ξ}})_{ξ<α} converges to f_{A} and since f_{B_{ξ}} ∈ Ũ for each ξ < α and Ũ is closed with respect to limits of transfinite sequences, we get f_{A} ∈ Ũ.

According to theorem known in general topology each T_{0}-space can be embedded in a space S^{β} for some infinite cardinal β. Note that each prime space is a T_{0}-space. So from this we get:

Theorem 3.4. Any topological space is a subspace of a pseudoradial space. Moreover, every T_{0}-space is a subspace of a pseudoradial T_{0}-space.

In order to extend this result to the class of T_{1}-spaces we cannot use the space S^{β}, because this space is not T_{1}. Let us recall that the cofinite topology on an underlying set X is the coarsest T_{1} topology on this set. Closed sets in the cofinite topology are finite sets and the whole set X.

For any cardinal number β, let (S^{β})_{1} be the topological space on the set {0, 1}^{β} with the topology which is the join of the product topology S^{β} and the cofinite topology on the set {0, 1}^{β}. If β is finite, then (S^{β})_{1} is discrete space.

If X is a T_{1}-space, then there exists an embedding e: X ↪ S^{β} of X into some topological power S^{β} of S. Since X is T_{1}, e: X ↪ (S^{β})_{1} is an embedding as well. So in order to prove that each T_{1}-space is a subspace of a T_{1}-pseudoradial space, it suffices to show that the space (S^{β})_{1} is pseudoradial. The proof of this result is only a bit more complicated than the proof of the fact that S^{β} is pseudoradial and we will omit it.

Theorem 3.6. Every T_{1}-space is a subspace of a pseudoradial T_{1}-space.

The same result doesn’t hold for Hausdorff spaces. A counterexample is any compact Hausdorff space X which is not pseudoradial. If X would be a subspace of a pseudoradial space, it would be itself pseudoradial (closed subspace of a pseudoradial space). We can take e.g. X = βω.

4 Coreflective subcategories with SA = Top

We showed that the category PsRad is an example of a coreflective subcategory of Top such that SA = Top. In what follows we find the smallest coreflective subcategory of Top with this property.

Theorem 4.1. Let A be a coreflective subcategory of Top. Then SA = Top if and only if S^{α} ∈ A for every infinite cardinal α.

Proof. Let A be a coreflective subcategory of Top for which SA = Top and α be any infinite cardinal. There exists a space X ∈ A such that S^{α} is a subspace of X. For each a ∈ α let p_{a}: S^{α} → S denote the a-th projection of topological power S^{α} onto S. The set (p_{a})^{−1}(0) is open in S^{α} so that there exists an open subset U_{a} in X such that U_{a} ∩ S^{α} = (p_{a})^{−1}(0). The map f_{a}: X → S given by f_{a}(x) = 0 for each x ∈ U_{a} and f_{a}(x) = 1 otherwise is a continuous extension of p_{a}: S^{α} → S. The map f: X → S^{α} with f_{a} = p_{a} ∘f for each a ∈ α is continuous and the restriction f|_{S^{α}} is the identity map on S^{α}. Hence f is a retraction and, consequently, f is a quotient map. Thus S^{α} ∈ A.

Conversely, if S^{α} belongs to A for any cardinal α, then any prime space belongs to SA and since SA is a coreflective subcategory of Top we obtain that SA = Top.

Corollary 4.3. A = CH({S^{α}; α is an infinite cardinal}) is the smallest coreflective subcategory of Top such that SA = Top.

We next present another class of generators of the category CH({S^{α}; α is an infinite cardinal}).

Let α be an infinite cardinal and B_{β} = {γ ∈ α ∪ {α}; γ ≥ β} for each β ∈ α. Then M(α) is the topological space on the set α∪{α} with the topology consisting of all B_{β}, β being a non-limit ordinal less then α or β = 0. These spaces have the following property:

Proposition 4.4. Let α be an infinite cardinal and M(α) be a subspace of X. Then there exists a retraction f: X → M(α).

Proof. For every non-limit ordinal β < α denote by U_{β} the union of all open subsets of X with U ∩ M(α) = B_{β} and put U_{0} = X. Clearly, if 0 ≤ β < β′ < α then U_{β} ⫌ U_{β′} and for each β < α U_{β} ∩ M(α) = B_{β}. Define f: X → M(α) by

f(x) = sup{β ∈ α : x ∈ U_{β}}.

Obviously, f^{−1}(B_{β}) = U_{β} for non-limit ordinal β. (x ∈ U_{β} ⇒ f(x) ≥ β ⇒ x ∈ f^{−1}(B_{β})), x ∉ U_{β} ⇒ f(x) < β ⇒ x ∉ f^{−1}(B_{β}).) Thus f is continuous. Moreover we have f(β) = β for β ∈ M(α) and f is a retraction.

Corollary 4.7. CH({M(α); α is a regular cardinal}) is the smallest coreflective subcategory of Top such that SA = Top.

Proof. If M(α) belongs to A, then according to [3, Proposition 3.5] the prime factor (M(α))_{α} = C(α) belongs to SA. Therefore SA contains PsRad = CH({C(α); α is a regular cardinal}) and, consequently, SA ⊇ SPsRad = Top.

On the other hand, if SA = Top, then there exists a topological space X such that M(α) is a subspace of X. According to Proposition 4.4 there exists a retraction f: X → M(α) and M(α) belongs to SA.

Using the spaces M(α) can be proved one more characterization of the smallest coreflective subcategory of Top such that SA = Top:

Proposition 4.9. Let A be a coreflective subcategory of Top. Then SA = Top if and only if for every regular cardinal α A contains a space X such that there exists x ∈ X with t(X, x) = α and for α = ω_{0} the prime factor X_{x} of X at x is, moreover, not finitely generated.

References

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Questions and notes

1. Which set-theoretical assumption have been used in [4] and [14]? The result of [14] is proved under 𝔭 = 𝔠, which is a consequence of MA. (𝔭=

minimal cardinality of an infinite subset of ω, which has SFIP and fails to have infinite pseudointersection. SFIP = strong finite intersection property = finite subfamilies have infinite intersection. Pseudointersection of a family = is almost included in each set of this family, P ⊂^{∗} F for each F ∈ F.) It is proved in [4] that CH implies that each ω ∪ {p} is a subspace of some regular pseudoradial space and there is a model in which this result doesn’t hold.

2. Using these results we can give a simple alternative proof of Kannan’s result that there is no roper subcategory of Top which is simultaneously coreflective and epireflective (i.e. closed with respect to the formation of topological sums, quotients, subspaces and products).

Let A be the smallest such category. If D is the 2-point discrete space, then D ∈ A and D^{ω} ∈ A. D^{ω} is not indiscretely generated, therefore S ∈ A and S^{α} ∈ A for each infinite cardinal α. Thus we get by Theorem 4.1 that A = SA = Top.

3. Let X be any topological space. Are there any bounds for cardinal invariants of a pseudoradial space Y such that X is a subspace of Y.

Radial character: In the paper [13] it is proved in fact that Gen(α) ⊆ SPsRad(2^{α}), which means that if t(X) ≤ α then there exists a pseudoradial space with σ_{C}(Y) ≤ 2^{α} such that X is a subspace of Y. For T_{0}- and T_{1}- spaces results of [13] yields only estimate σ_{C}(Y) ≤ 2^{card X}. [6, Example 3.1] ℕ_{p} = ω ∪ {p} is a countable space, which is not subsequential ⇒ σ_{C}(Y) > ℵ_{0}.

4. Are pseudoradial spaces precisely the spaces determined by nets which contain cofinal well-ordered subnet?

5. Can we replace T_{1} with sober spaces in Theorem 3.6?