kimjuagt.dvi @ Applied General Topology c© Universidad Politécnica de Valencia Volume 5, No. 1, 2004 pp. 1- 10 Iterated starcompact topological spaces Junhui Kim Abstract. Let P be a topological property. A space X is said to be k-P-starcompact if for every open cover U of X, there is a subspace A ⊆ X with P such that stk(A, U) = X. In this paper, we consider k-P- starcompactness for some special properties P and discuss relationships among them. 2000 AMS Classification: 54D20, 54B05. Keywords: countably compact, n-starcompact, (n, k)-starcompact, pseudo- compact. 1. Introduction Let X be a topological space and U a collection of subsets of X. For Ø 6= A ⊆ X, let st(A, U) = st1(A, U) = ⋃ {U ∈ U : A ∩ U 6= Ø} and stn+1(A, U) = st(stn(A, U), U) for all n ∈ N. We simply write stn(x, U) for stn({x}, U). A space X is called n-starcompact (n1 2 -starcompact) [6] if for every open cover U of X there is a finite subset F of X (finite subcollection V of U) such that stn(F, U) = X (stn( ⋃ V, U) = X). Let Ñ = N ∪ {n1 2 : n ∈ N}. By definition, every n-starcompact space is (n + 1 2 )-starcompact for n ∈ Ñ. It is known that 1-starcompactness is equivalent to countable compactness for Hausdorff spaces. Moreover, every n-starcompact regular space is 21 2 -starcompact for n ≥ 3, n ∈ Ñ, and 21 2 -starcompactness is equivalent to pseudocompactness for Tychonoff spaces [1]. Behaviours of the above mentioned star-covering properties were studied in [1, 6, 7]. By replacing ‘finite’ with ‘countable’ in the definition, n-starcompactness was extended to n-star-Lindelöffness in [1]. As we have seen, finiteness plays an important role in the concept of n-starcompactness. In what follows, we may replace finiteness with some topological properties to get some new con- cepts. Given a topological property P, a space X is called k-P-starcompact if for every open cover U of X, there is a subspace A ⊆ X with P such that stk(A, U) = X. Ikenaga [4] and Song [7] considered 1-P-starcompactness for 2 Junhui Kim P being compact. We are especially interested in k-P-starcompact spaces for P being n-starcompact, and call them iterated starcompact spaces in general. More precisely, a space X is said to be (n, k)-starcompact if for every open cover U of X there is an n-starcompact subspace A of X such that stk(A, U) = X. For the sake of unification, a compact space is called 1 2 -starcompact. In fact, the above definitions appeared in [6] but no further investigation has been done so far. By definition, we have the following lemma. Lemma 1.1. (i) Every (n, k)-starcompact space is (n + k)-starcompact for n ∈ Ñ and k ∈ N. (ii) Every (n1, k)-starcompact space is (n2, k)-starcompact for n1, n2 ∈ Ñ with n1 ≤ n2 and k ∈ N. (iii) Every (n, k1)-starcompact space is (n, k2)-starcompact for n ∈ Ñ and k1, k2 ∈ N with k1 ≤ k2. By applying known properties, we obtain Diagram 1 in the class of regular spaces. For convenience, (n, k)-starcompactness is abbreviated as stn,k. st1 st2 compact st 1 2 st 1 2 ,1 st 1 2 ,2 countably compact st1 st1,1 st1,2 st1 1 2 st1 1 2 ,1 st1 1 2 ,2 st2 st2,1 st2,2 st2 1 2 � locally compact - - - - - - - - - - ? ? ? ? ? ? ? ? ? ? ? � � � � � � ��� � � � � � � ��� Diagram 1 (In the class of regular spaces) In Section 2, we provide examples to distinguish iterated starcompact prop- erties around (1, 1)-starcompactness and consider their relations with other covering properties. Section 3 is devoted to distinguish properties weaker than 2-starcompactness. Throughout this paper, ω (ω1) is the first infinite (uncount- able) cardinal and c is the continuum. For any set A, the cardinality of A is denoted by |A|. Undefined concepts and symbols can be found in [2]. Iterated starcompact topological spaces 3 2. (1, 1)-starcompact Spaces A space X is said to be L-starcompact if for every open cover U of X there exists a Lindelöf subspace L such that st(L, U) = X. By definition, every (1 2 , 1)-starcompact space is L-starcompact, (1, 1)-starcompact and 11 2 - starcompact; every 11 2 -starcompact space is both (11 2 , 1)-starcompact and 2- starcompact; and every (1, 1)-starcompact space is both 2-starcompact and (11 2 , 1)-starcompact. These relationships can be described in the following di- agam. � ��3 - Q Q Qs - - Z Z Z~� � �> (1 2 , 1)-starcompact (1, 1)-starcompact L-starcompact 11 2 -starcompact (11 2 , 1)-starcompact 2-starcompact Diagram 2 In this section, we shall first provide some examples to show the difference among concepts in Diagram 2. Lemma 2.1. [6] If a regular space X contains a closed discrete subspace Y such that |Y | = |X| ≥ ω, then X is not 11 2 -starcompact. Example 2.2. There is a 2-starcompact, L-starcompact Tychonoff space which is not (11 2 , 1)-starcompact. Let R be a maximal almost disjoint family of infinite subsets of ω with |R| = c. It is proved that the Isbell-Mrówka space Ψ = ω ∪R is 2-starcompact in [1]. Since Ψ is separable, it is L-starcompact. Note that every 11 2 -starcompact subspace of Ψ is compact. For, if there exists a 11 2 - starcompact non-compact subspace X ⊆ Ψ, then |X ∩ R| < |X| ≤ ω by Lemma 2.1. It follows from |X ∩ R| = |{R1, · · · , Rn}| < ω that there exists A ⊆ X ∩ ω such that |A| = ω and A ∩ ⋃n i=1 Ri = Ø. This implies that X is not pseudocompact, which is a contradiction. Enumerate R as {Rβ : β < c}. Since the intersection of every compact subspace of Ψ with R is finite, we can enumerate all compact subsets of Ψ as K = {Fα : α < c}. For each α < c, choose βα > α such that |Rβα ∩ Fα| < ω. In addition, we may requre βα < βα′ whenever α < α′. Choose an open neighborhood O(Rβα) of Rβα such that O(Rβα) ∩ Fα = Ø. Let I = {βα : α < c}. Then U = {{n} : n ∈ ω} ∪ {{Rα} ∪ Rα : α ∈ c r I} ∪ {O(Rβα) : α ∈ c} is an open cover of Ψ. Let K be any compact subspace of Ψ. Then K = Fα for some α < c. By the construction of U, Rβα 6∈ st(K, U). Therefore, Ψ is not (11 2 , 1)-starcompact. � Example 2.3. There is a (1, 1)-starcompact Tychonoff space which is neither 11 2 -starcompact nor L-starcompact. Let τ be a regular cardinal with τ ≥ ω1. Let D be the discrete space with |D| = τ and let D∗ = D∪{∞} be the one-point 4 Junhui Kim compactification of D. Consider X = (D∗ × (τ + 1)) r {〈∞, τ〉} as a subspace of the usual product space D∗ × (τ + 1). Since D∗ × τ is a countably compact dense subspace of X, X is (1, 1)-starcompact. But X is not 11 2 -starcompact, since |X| = |D| and D × {τ} is a closed discrete subspace of X. Now, we will show that X is not L-starcompact. Enumerate D as {dα : α < τ}. For each α < τ, choose an open set Uα = {dα} × (α, τ]. Then U = {Uα : α < τ} ∪ {D ∗ × τ} is an open cover of X. Let L be any Lindelöf subspace of X. Then L∩(D×{τ}) must be countable. Let L′ = Lr ⋃ {L∩Uα : 〈dα, τ〉 ∈ L ∩ (D × {τ})}. Without loss of generality, we may assume L ′ 6= Ø. Since L′ is closed in L, L′ is Lindelöf. Note that L′ ⊆ D∗×τ. Let π : D∗×τ → τ be the projection. Then π(L′) is a Lindelöf subspace of the countably compact space τ. Therefore there exists κ0 < τ which is greater than all elements of π(L′), i.e., Uα ∩ L ′ = Ø for all α ≥ κ0. Since τ is a regular cardinal with τ ≥ ω1 and L ∩ (D × {τ}) is countable, there exists some κ < τ such that κ0 < κ and Uκ ∩ L = Ø. Because Uκ is the only one element of U containing dκ, dκ 6∈ st(L, U). Therefore, X is not L-starcompact. � Example 2.4. There is an L-starcompact and (1, 1)-starcompact Tychonoff space X which is not 11 2 -starcompact. Let Ψ = ω ∪ R be the Isbell-Mrówka space, where R is a maximal almost disjoint family of infinite subsets of ω with |R| = c and let D be the discrete space such that |D| = |R| and D ∩ R = Ø. Let Y = (D∗ × (ω1 + 1)) r {〈∞, ω1〉}, where D ∗ = D ∪ {∞} is the one-point compactification of D. Take a bijection i : R → D×{ω1}. Let X be a quotient space of Ψ ∪ Y and π : Ψ ∪ Y → X a quotient mapping which identifies r with i(r) for each r ∈ R. Then X = π(ω) ∪ π(Y ) = π(Ψ) ∪ π(D∗ × ω1). Since X is locally compact Hausdorff, it is Tychonoff. First, we will show that X is (1, 1)-starcompact. Let U be an open cover of X. Note that A = π(D∗ × ω1) is a countably compact dense subset of π(Y ). Hence π(Y ) ⊆ st(A, U). Since π(ω) is relatively countably compact in π(Ψ), B = ω r st(A, U) is finite. Thus st(A ∪ B, U) = X and A ∪ B is a countably compact subspace of X. Now, we will show that X is L-starcompact. Since A is countably compact, there is a finite subset F of A such that A ⊆ st(F, U). Moreover, π(ω) is a countable dense subset of π(Ψ), thus we have st(F∪π(ω), U) = X and F∪π(ω) is a Lindelöf subspace of X. But π(R) is closed and discrete in X and |π(R)| = |X|. Therefore, X is not 11 2 -starcompact. � Example 2.5. There is a 11 2 -starcompact, L-starcompact Hausdorff space which is not (1, 1)-starcompact. Let X = [0, 1] and let τ0 be the Euclidean topology on X. Define τ1 = {U r F : U ∈ τ0, F is a countable subset of X}. Then (X, τ1) is Hausdorff. We will show that (X, τ1) is 1 1 2 -starcompact. Let U be a basic open cover of (X, τ1). For each U ∈ U, select an open subset V (U) of (X, τ0) and a countable subset F(U) of X such that U = V (U) r F(U). Then V = {V (U) : U ∈ U} is an open cover of (X, τ0). Since (X, τ0) is com- pact, V has a finite subcover V0. Let U0 = {U ∈ U : V (U) ∈ V0}. Then |X r ⋃ U0| ≤ ω. Since every neighborhood of each point of X r ⋃ U0 meets⋃ U0, st( ⋃ U0, U) = X. It is easy to prove that (X, τ1) is Lindelöf. Note that Iterated starcompact topological spaces 5 every countable subset is closed and discrete in (X, τ1). So every countably compact subspace is finite. Since (X, τ1) is not countably compact (i.e., not 1-starcompact), it is not (1, 1)-starcompact. � A space X is said to be meta-Lindelöf (para-Lindelöf ) if every open cover of X has a point (locally) countable open refinement. It is well-known that every pseudocompact para-Lindelöf Tychonoff space is compact. Theorem 2.6. Let X be a meta-Lindelöf T1 space. If X is (1, 1)-starcompact, then it is 11 2 -starcompact. Proof. Let U be an open cover of X. Since X is meta-Lindelöf, we may as- sume that U is point countable. Since X is (1, 1)-starcompact, there exists a countably compact subspace A of X such that st(A, U) = X. We may assume A ∩ U 6= Ø for all U ∈ U. Now, we will show that some finite subcollection V of U covers A. Therefore st( ⋃ V, U) = X. Suppose that it is not true, and pick an arbitrary point x0 ∈ A. Denote by Vx0 the subcollection {V ∈ U : x0 ∈ V } of U. Since A is countably compact and Vx0 is countable, A r ⋃ Vx0 6= Ø (Otherwise, we have A ⊆ ⋃ Vx0, and thus there exists a finite subfamily of Vx0 which covers A). Inductively, we can choose an infinite sequence {xn : n ∈ ω} such that xn ∈ A r ⋃ i