Irreducibility of product spaces with finitely many points removed

Paul B. Larson^{∗}

August 1, 2005

Abstract We prove an induction step that can be used to show that in certain cases the removal of finitely many points from a product space produces an irreducible space. For example, we show that whenever γ is less than ℵ_{ω}, removing finitely many points from the product of γ many first countable compact spaces gives an irreducible space. This result answers questions asked privately by Alexander Arhangel’skii.

If O is a collection of open sets corresponding to some topological space, we call O′ an open refinement of O if every member of O′ is an open set and a subset of a member of O. A collection S of sets is said to be minimal if for each S ∈ S there is an x ∈ S which is not in any other member of S. A topological space is said to be irreducible if every open cover of the space has a minimal open refinement covering the space (see [1, 2, 3, 4]). In this note we prove an induction step which can be used to show in certain cases that the removal of finitely many points from a product space yields an irreducible space. This induction step applies to regular limits, and the most immediate question left open is how to get past singular ones. One larger project, as we understand it, is to collect examples of noncompact irreducible spaces, though we cannot claim expertise on this topic.

A cube in a product space X = ∏_{a∈A} X_{a} is a set of the form

I_{f}^{A} = {x ∈ X | ∀a ∈ dom(f) x(a) ∈ f(a)}

for some finite function f with domain contained in A such that each f(a) is an open subset of X_{a}. We call the domain of f the support of I_{f}^{A}. We say that a subspace Y of a product space X is cube-irreducible if every cover of Y by open subsets of X has a minimal open refinement covering Y and consisting of cubes. Note that every compact subspace of a product space is cube-irreducible.

Following [5], we say that a topological space X is (γ, ∞)-compact if every open cover of X has a subcover of cardinality less than γ.

∗The work in this paper was supported in part by NSF Grant DMS-0401603. The author would also like to thank Alexander Arhangel’skii, Dennis Burke and Sheldon Davis for introducing him to this topic.

Theorem 0.1 below gives an argument that applies to the removal of one point from a product space, and Theorem 0.2 gives a slightly generalized argument that handles the removal of finitely many points. One could naturally ask about the removal of infinite sets satisfying certain conditions.

Note that condition (1) in Theorem 0.1 and Theorem 0.2 are satisfied when each X_{α} is compact and first countable, since then the removal of finitely many points from X_{α} produces a Lindelöf space, and the product of a Lindelöf space with a compact space is Lindelöf.

Theorem 0.1. Let κ be an uncountable regular cardinal. For each α < κ let X_{α} be a topological space and let e_{α} be a point in X_{α}. For each A ⊂ κ, let X(A) denote ∏_{α∈A} X_{α}, and let e(A) denote the element of X(A) whose α-th coordinate is e_{α} for each α ∈ A. Suppose that the following hold:

1. for all α < κ, (X_{α} \ {e_{α}}) × X(κ \ (α + 1)) is (κ, ∞)-compact;

2. for all β < γ < κ, X([β, γ)) \ {e([β, γ))} is irreducible.

Then X(κ) \ {e(κ)} is irreducible. Furthermore, if for all β < γ < κ,

X([β, γ)) \ {e([β, γ))}

is cube-irreducible, then so is X(κ) \ {e(κ)}.

Proof. Since every open subset of X(κ) \ {e(κ)} is a union of cubes in X(κ) not containing e(κ), it suffices to consider the case where O is a set of cubes in X(κ) not containing e(κ) which covers X(κ) \ {e(κ)}. It follows that for every I_{f}^{κ} ∈ O, e_{α} ∉ f(α) for some α ∈ dom(f). For each α < κ, let E_{α} be a subset of O of cardinality less than κ covering the set of z ∈ X(κ) for which α is the least β < κ such that z(β) ≠ e_{β}. We may assume that whenever f is a function defining a cube in E_{α} and β ∈ dom(f) ∩ α, then e_{β} ∈ f(β). Since for each α < κ the union of the supports of the members of E_{α} is a bounded subset of κ, and since κ is regular, we may fix a club subset C of κ with 0 ∈ C such that for each β ∈ C and each α < β, the support of each member of E_{α} is contained in β. For each β ∈ C, let Y_{β} be the set of z ∈ X(κ) for which the least α such that z(α) ≠ e_{α} is in the interval [β, c(β)), where c(β) is defined to be the least element of C above β.

Then for each β ∈ C, O_{β} = ⋃{E_{α} | α ∈ [β, c(β))} is a covering of Y_{β} and {I_{f↾[β,c(β))}^{[β,c(β))} | I_{f}^{κ} ∈ O_{β}} is a covering of X([β, c(β)))\{e([β, c(β)))}, which can be refined to a minimal open covering U_{β} of X([β, c(β))) \ {e([β, c(β)))}. For each open set O in each U_{β}, pick a function f such that I_{f}^{κ} ∈ O_{β} and O ⊂ I_{f↾[β,c(β))}^{[β,c(β))} and let F (O) be the set of z ∈ X(κ) such that z↾[β, c(β)) ∈ O and z ∈ I_{f}^{κ}. Note that for such an f, e_{α} ∈ f(α) for every α ∈ dom(f) ∩ β, and also that if O is a cube then so if F (O).

Then O^{∗} = {F (O) : O ∈ ⋃{U_{β} | β ∈ C}} is an open refinement of O. It remains to see that O^{∗} covers X(κ) \ {e(κ)} and is minimal.

To see that O^{∗} covers X(κ) \ {e(κ)}, fix z ∈ X(κ) \ {e(κ)} and let β be the least γ ∈ C such that z(α) ≠ e_{α} for some α in the interval [γ, c(γ)). Then z↾[β, c(β)) is in some member O of U_{β}, and since z(α) = e_{α} for all α < β, z is in F (O).

To see that O^{∗} is minimal, fix β < κ and O ∈ U_{β}. Since U_{β} is a minimal cover of X([β, c(β))) \ {e([β, c(β)))}, there exists a z_{0} ∈ X([β, c(β))) \ {e([β, c(β)))} which is not an element of any member of U_{β} other than O. Let z ∈ X(κ) be the extension of z_{0} taking value e_{α} at each α outside of [β, c(β)). Then z is not a member of any F (O′) for any O′ ≠ O in U_{β}, and for each β′ ∈ κ \ {β} and each O′ ∈ U_{β′}, z is not in F (O′), since every element of O′ takes value other than e_{α} at some point α ∈ [β′, c(β′)).

If s is a function with domain an ordinal, we let l(s) denote the domain (equivalently, the length) of s.

Theorem 0.2. Let κ be an uncountable regular cardinal. For each α < κ let X_{α} be a Hausdorff topological space. For each A ⊂ κ, let X(A) denote ∏_{α∈A} X_{α}. Let E = {e_{i} : i < n} be a subset of X(κ) and for each A ⊂ κ let E(A) denote {e_{i}↾A | i < n}.

Let Σ denote {e_{i}↾γ | i < n ∧ γ < κ}, and for each s ∈ Σ let N(s) denote the set of x ∈ X_{l(s)} such that s^{⌢}⟨x⟩ ∈ Σ. For each s ∈ Σ and each γ ∈ (l(s), κ), let M(s, γ) denote

{x↾[l(s), γ)) | x ∈ X(γ) ∩ Σ ∧ s ⊂ x}.

Suppose that the following hold:

1. for all s ∈ Σ,

(X_{l(s)} \ N(s)) × X(κ \ (l(s) + 1))

is (κ, ∞)-compact;

2. for all s ∈ Σ and all γ ∈ (l(s), κ), X([l(s), γ)) \ M(s, γ) is irreducible.

Then X(κ) \ E is irreducible. Furthermore, if for all s ∈ Σ and all γ ∈ [l(s), κ), X([l(s), γ)) \ M(s, γ) is cube-irreducible, then so is X(κ) \ E.

Proof. Fix an open covering O of X(κ)\E. Since every open subset of X(κ)\E is a union of cubes disjoint from E, it suffices to consider the case where O is a set of cubes disjoint from E covering X(κ) \ E. Then for every I_{f}^{κ} ∈ O and each e ∈ E there is a β ∈ dom(f) such that e(β) ∉ f(β). Let D be a finite subset of κ such that for each pair e_{i}, e_{j} of distinct elements of E there is an α ∈ D such that e_{i}(α) ≠ e_{j}(α). For each α in D, fix disjoint open subsets O(i, α) (i < n) of X_{α} containing e_{i}(α) but no e_{j}(α), j ≠ i. We may assume furthermore that for each function f with I_{f}^{κ} ∈ O, D ⊂ dom(f) and, for each α ∈ D and i < n, if e_{i}(α) ∈ f(α) then f(α) ⊂ O(i, α).

For each s ∈ Σ, let V_{s} be a subset of O of cardinality less than κ covering the set of z ∈ X(κ) such that s ⊂ z and for which l(s) is the least β < κ such that z↾β ∉ Σ (note that this set, and thus V_{s}, might be empty). We may assume that whenever f is a function defining a cube in V_{s} and β ∈ dom(f) ∩ l(s), s(β) ∈ f(β). For each α < κ, the union of the supports of the members of all the V_{s} with l(s) < α is a bounded subset of κ, and so we may fix a club subset C of κ satisfying the following:

• 0 ∈ C;

• C ∩ ((max(D) + 1) \ {0}) = ∅;

• each nonzero element of C is a limit ordinal;

• for each β ∈ C and each s ∈ Σ with l(s) < β, the support of each member of V_{s} is contained in β.

For each s ∈ Σ with l(s) ∈ C, let Y_{s} be the set of z ∈ X(κ) such that s ⊂ z and for which the least α such that z↾α ∉ Σ is in the interval (l(s), c(l(s))), where for each β < ω_{1}, c(β) is defined to be the least element of C above β.

Then for each s ∈ Σ with l(s) ∈ C,

O_{s} = ⋃{V_{t} | s ⊂ t ∧ t ∈ Σ ∧ l(t) ∈ [l(s), c(l(s)))}

is a covering of Y_{s} and

{I_{f↾[l(s),c(l(s)))}^{[l(s),c(l(s)))} | I_{f}^{κ} ∈ O_{s}}

is a covering of X([l(s), c(l(s))))\M(s, c(l(s))), which can be refined to a minimal open covering U_{s} of X([l(s), c(l(s))))\M(s, c(l(s))). For each open set O in each U_{s}, pick a function f such that I_{f}^{κ} ∈ O_{s} and O ⊂ I_{f↾[l(s),c(l(s)))}^{[l(s),c(l(s)))} and let F (O) be the set of z ∈ X(κ) such that z↾[l(s), c(l(s))) ∈ O and z ∈ I_{f}^{κ}. Note that for such an f, s(α) ∈ f(α) for every α ∈ dom(f) ∩ l(s). Note also that if O is a cube then so if F (O).

Then O^{∗} = {F (O) : O ∈ ⋃{U_{s} | s ∈ Σ ∧ l(s) ∈ C}} is an open refinement of O. It remains to see that O^{∗} covers X(κ) \ E and is minimal.

To see that O^{∗} covers X(κ) \ E, fix z ∈ X(κ) \ E and let β be the least γ ∈ C such that z↾α ∉ Σ for some α in the interval (γ, c(γ)). Then z↾β ∈ Σ, z↾[β, c(β)) is in some member O of U_{z↾β}, and z is in F (O), since for each α < β and each I_{f}^{κ} ∈ O_{z↾β}, z(α) ∈ f(α).

To see that O^{∗} is minimal, fix s ∈ Σ with l(s) ∈ C and fix O ∈ U_{s}. Since U_{s} is a minimal cover of X([l(s), c(l(s)))) \ M(s, c(l(s))), there exists a z_{0} ∈ X([l(s), c(l(s)))) \ M(s, c(l(s))) which is not an element of any member of U_{s} other than O. Fix i < n as follows. If l(s) = 0, let i be the unique j < n such that z_{0}(α) ∈ O(j, α) for all α ∈ D, if such a j exists, and if no such j exists let i = 0. If l(s) > 0, let i be the unique j such that s ⊂ e_{j}. Let z = z_{0} ∪ (e_{i}↾(κ \ [l(s), c(l(s))). Then z ∈ X(κ) \ E, and z is not a member of any F (O′) for any O′ ≠ O in U_{s}.

Suppose that l(s) ≠ 0 and fix s′ ∈ Σ such that l(s′) ∈ C and s′↾D ≠ s↾D. Fix α ∈ D such that s(α) ≠ s′(α). Since s(α) ∉ f′(α) for all functions f′ with I_{f′}^{κ} ∈ O_{s′}, z is not in F (O′) for any O′ ∈ U_{s′}.

Now, suppose that s′ ∈ Σ, l(s′) ∈ C \ {l(s)} and s′↾D = s↾D or l(s) = 0. Then s′ is an initial segment of s or vice-versa. First, suppose that s′ is an initial segment of e_{i} (which subsumes the subcase s′ ⊂ s). Then e_{i}↾[l(s′), c(l(s′))) ∉ O′ for any O′ ∈ U_{s′}, and so z ∉ F (O′), since [l(s′), c(l(s′))) ⊂ (κ \ [l(s), c(l(s)))).

Finally, if s′ is not an initial segment of e_{i} (in this case l(s) = 0, though we don’t use this fact), then s′ is an initial segment of some e_{j} and for some α ∈ D, z(α) is not in O(j, α). Then every F (O′) for some O′ ∈ U_{s′} is a subset of I_{f′}^{κ} for a function f′ with the property that O(j, α) ⊂ f′(α), and so z ∉ F (O′).

0.3 Remark. One could weaken condition (2) in the statement of Theorem 0.2 to say that (a) for all γ < κ, X([0, γ)) \ M(∅, γ) is irreducible, and (b) there exists a ρ < κ such that for all s ∈ Σ of length greater than ρ and for all γ ∈ (l(s), κ), X([l(s), γ)) \ M(s, γ) is irreducible. Then in the proof, make the least nonzero element of C greater than ρ and max(D). One has in this case that M(s, γ) is a singleton for each nonempty s ∈ Σ with l(s) ∈ C.

Naturally, one would like to improve the proofs of Theorem 0.1 and Theorem 0.2 to get past singular limits. Another natural issue is to find base cases for the induction steps proved above. The following fact follows easily from standard arguments.

Theorem 0.4. The removal of finitely many points from a countable product of first countable compact spaces produces a cube-irreducible space.

Adding this to Theorem 0.2, one gets the following.

Corollary 0.5. The removal of finitely many points from a product of less than ℵ_{ω} many first countable compact spaces produces a cube-irreducible space.

References

[1] R. Arens, J. Dugundji, Remark on the concept of compactness, Portugal. Math. 9 (1950) 141-143

[2] J.R. Boone, On irreducible spaces, Bull. Austral. Math. Soc. 12 (1975), 143–148

[3] J.R. Boone, On irreducible spaces. II, Pacific J. Math. 62 (1976), no. 2, 351–357

[4] E.K. van Douwen, W.F. Pfeffer, Some properties of the Sorgenfrey line and related spaces, Pacific J. Math. 81 (1979), no. 2, 371–377

[5] J.E. Vaughan, Countably compact and sequentially compact spaces, Handbook of set-theoretic topology, K. Kunen, J.E. Vaughan, eds., North-Holland, Amsterdam, 1984, 569–602

Department of Mathematics and Statistics Miami University, Oxford, Ohio 45056, United States larsonpb@muohio.edu