Two cardinal inequalities for functionally Hausdorff spaces
Alessandro Fedeli
Abstract. In this paper, two cardinal inequalities for functionally Hausdorff spaces are established. A bound on the cardinality of theτ θ-closed hull of a subset of a functionally Hausdorff space is given. Moreover, the following theorem is proved: ifXis a functionally Hausdorff space, then|X| ≤2χ(X)wcd(X).
Keywords: cardinal functions,τ θ-closed sets,w-compactness degree Classification: 54A25, 54D20
A spaceX is said to be functionally Hausdorff if whenever x6=y in X there is a continuous real valued function f defined on X such that f(x) = 0 and f(y) = 1. A well-known Arkhangel’skii’s theorem states that ifX is a Hausdorff space, then|X| ≤2χ(X)L(X)([1], [6]). Bella and Cammaroto [2] established some cardinal inequalities for Urysohn spaces that improve, for non regular spaces, the Arkhangel’skii’s formula. In this paper, a bound on the cardinality of the τ θ-closed hull of a subset of a functionally Hausdorff space and a bound on the cardinality of a functionally Hausdorff space are given. We refer the reader to [3]
and [4] for notations and definitions not explicitly given. All topological spaces considered here are assumed to be infinite. Let E be a set; the cardinality of E is denoted by |E|, Pk(E) is the collection of all subsets of E of cardinality
≤k. χ(X) and L(X) denote respectively the character and the Lindel¨of degree of a spaceX.
Definition 1 [5]. Let A be a subset of a space X. A is called τ-open if A is a union of cozero-sets of X. The τ-closure of A, denoted by clτ(A), is the set of all points x ∈ X such that any cozero-set neighbourhood of x intersects A.
The τ-interior of A, denoted by intτ(A), is the set of all x such that there is a cozero-set neighbourhood ofxcontained inA.
Definition 2. LetX be a topological space andAa subset ofX. Theτ θ-closure ofA, denoted byclτ θ(A), is the set of all pointsx∈X such thatclτ(V)∩A6=∅ for every open neighbourhoodV of x. Ais said to beτ θ-closed ifA= clτ θ(A).
As pointed to me by S. Watson, theτ θ-closure is not in general idempotent.
Definition 3. LetX be a topological space andAa subset ofX. Theτ θ-closed hull ofA, denoted by[A]τ θ, is the smallest τ θ-closed subset ofX containingA.
Clearly, [A]τ θ = T
{F : A ⊂ F and clτ θ(F) = F}. For every space X and everyA ⊂X we haveA ⊂clτ θ(A) ⊂[A]τ θ ⊂clτ(A). It is obvious that ifX is a Tychonoff space, thenA= clτ θ(A) = [A]τ θ = clτ(A) for anyA⊂X.
The next result gives some conditions on a functionally Hausdorff space which are equivalent to clτ θ= clτ.
Proposition 4. For a functionally Hausdorff spaceX the following conditions are equivalent:
(i) For eachτ-open setV ofX,V = clτ(V).
(ii) For each open setGofX,G⊂intτ(clτ(G)).
(iii) For each subset AofX,clτ θ(A) = clτ(A).
(iv) For eachτ-open subsetV ofX,clτ θ(V) = clτ(V).
Proof: (i)⇔(ii) Lemma 28 in [9]. (ii)⇒(iii) LetA⊂X andx /∈clτ θ(A), then there is an open neighbourhoodGofxsuch that clτ(G)∩A=∅. By hypothesis G ⊂ intτ(clτ(G)), then there is a cozero set V such that x ∈ V ⊂ clτ(G), so V ∩A = ∅ and x /∈ clτ(A). Hence, clτ θ(A) = clτ(A). (iii) ⇒ (iv) is obvious.
(iv)⇒(i) LetV be aτ-open subset of X, by hypothesis clτ θ(V) = clτ(V). Now let x /∈ V, then there is an open set G such that x ∈ G and G∩V = clτ(V).
Since V is τ-open, we have clτ(G)∩V = ∅, hence x /∈ clτ θ(V). Therefore,
V = clτ θ(V) = clτ(V).
Remark 5. A functionally Hausdorff spaceX is called weakly absolutely closed [8] provided that everyτ-open filter base on X has an adherent point. An SW space is a functionally Hausdorff spaceX such that every point-separating subal- gebra ofC∗(X) which contains the constants is uniformly dense inC∗(X) [8]. It is worth noting that by Lemma 25 in [9] and Proposition 4, a functionally Hausdorff space X is weakly absolutely closed iff it is an SW space and clτ θ(A) = clτ(A) for everyA⊂X.
The following result gives an upper bound on theτ θ-closed hull.
Theorem 6. Let X be a functionally Hausdorff space. If A is a subset of X, then|[A]τ θ| ≤ |A|χ(X).
Proof: Letm=χ(X) andk=|A|. For eachx∈X letB(x) be a base forX at the pointxsuch that |B(x)| ≤ m. Ifx∈ clτ θ(A), choose a point in clτ(U)∩A for every U ∈ B(x) and let Bx be the set so obtained. Clearly, x ∈ clτ θ(Bx) and |Bx| ≤m. Let Gx ={clτ(U)∩Bx :U ∈ B(x)}. For every U ∈ Bx we have x∈clτ θ(clτ(U)∩Bx), in fact, ifV ∈ B(x) letW ∈ B(x) such thatW ⊂V ∩U, then
∅ 6= clτ(W)∩Bx⊂clτ(V ∩U)∩Bx ⊂clτ(V)∩(clτ(U)∩Bx).
SinceX is functionally Hausdorff, thenT
{clτ θ(clτ(U)∩Bx) :U ∈ B(x)}={x}, in fact lety6=x, then there exist open setsGandH such thatx∈G,y∈H and clτ(G)∩clτ(H) =∅, now letU ∈ B(x) such thatU ⊂G, then clτ(H)∩clτ(U) =∅, so y /∈ T{clτ θ(clτ(U) : U ∈ B(x))}, and, a fortiori, y /∈ T{clτ θ(clτ(U)∩Bx) :
U ∈ B(x)}. So the map ψ : clτ θ(A) → Pm(Pm(A)) defined by ψ(x) = Gx for every x ∈ clτ θ(A), is one to one. Since |Pm(Pm(A))| ≤ (km)m = km, then
|clτ θ(A)| ≤ km = |A|χ(X). Let A0 = A and, by transfinite induction, define for every α < m+ sets Aα such that Aα = clτ θ(S
{Aβ : β < α}). Clearly S{Aα : α < m+} ⊂ [A]τ θ. Now let x ∈ clτ θ(S
{Aα : α < m+}), for each V ∈ B(x) choose a point in clτ(V)∩(S
{Aα :α < m+}) and letB be the set so obtained, obviously B ∈ Pm(S
{Aα :α < m+}) and x∈ clτ θ(B). Since m+ is regular, there is an ordinalα < m+ such thatB⊂Aα, so
x∈clτ θ(B)⊂clτ θ(Aα)⊂Aα+1⊂[
{Aα:α < m+},
therefore S{Aα : α < m+} is τ θ-closed. Hence [A]τ θ = S{Aα : α < m+}.
It remains to show that |Aα| ≤ km for each α < m+ (this is equivalent to
|S
{Aα : α < m+}| ≤ km). Suppose there is an ordinal α < m+ such that
|Aα|> km and let γ= min{α:|Aα| > km}. Since |Aα| ≤km for everyβ < γ, we have|S
{Aβ :β < γ}| ≤km. NowAγ = clτ θ(S
{Aβ :β < γ}), hence
|Aγ|=|clτ θ([
{Aβ :β < γ})| ≤ |[
{Aβ :β < γ}|χ(X)≤(km)m=km,
a contradiction.
Definition 7. LetX be a topological space. The w-compactness degree ofX, denoted bywcd(X), is defined as the smallest infinite cardinal numberkwith the property that for every open coverU ofX there is a subcollectionV ∈ Pk(U)for whichX=S
{clτ(V) :V ∈ V}.
For every spaceX we havewcd(X)≤L(X) and this inequality can be proper.
Example 8. Let X be any infinite T3-space such that every continuous real valued function defined onX is constant. Clearlywcd(X) =ℵ0< L(X).
Example 9. For eachα < ω1 letI(α) ={α}×an open interval in the real line.
SetX =ω1∪S
{I(α) :α < ω1}and forx, y∈X definex < yif (i)x, y∈ω1 and x < yin ω1, or (ii) x∈ω1,y ∈I(β) andx≤β in ω1, or (iii) x∈I(γ),y∈ω1 andγ < y inω1, or (iv)x∈I(α), y∈I(β) andα < β in ω1, or (v)x, y ∈I(α) andx < yin I(α). Letσbe the order topology onX. LetY =X∪ {ω1}, define x < ω1for everyx∈X and let̺be the order topology onY. Ifτ is the topology onY generated by̺∪ {Y −L:L is the set of limit ordinals inY − {ω1}}, then (Y, τ) is a functionally HausdorffH-closed space which fails to be Lindel¨of [7], so wcd(Y) =ℵ0< L(Y).
Theorem 10. IfX is a functionally Hausdorff space, then|X| ≤2χ(X)wcd(X). Proof: Letm=χ(X)wcd(X) and for everyx∈X letB(x) be a base forX at the pointxsuch that |B(x)| ≤m. Construct a family {Cα:α < m+}of subsets ofX such that
(1) for anyα < m+ Cα isτ θ-closed;
(2) for anyα < m+ |Cα| ≤2m; (3) ifα < β < m+, thenCα⊂Cβ; (4) for any α < m+, if U ⊂ S
{B(x) : x∈ S
{Cβ : β < α}}, |U| ≤ m and X−S
{clτ(U) :U ∈ U} 6=∅, thenCα−S
{clτ(U) :U ∈ U} 6=∅.
The construction is done by transfinite induction. Letp∈X andC0 ={p}.
Let 0 < α < m+ and assume that Cβ has been constructed for every β < α.
Let Bα = S{B(x) : x ∈ S{Cβ : β < α}}, clearly |Bα| ≤ 2m. For any U ⊂ Bα such that |U| ≤ m and X −S
{clτ(U) : U ∈ U} 6= ∅, choose a point in X−S
{clτ(U) : U ∈ U} and letA be the set so obtained, obviously|A| ≤2m. LetCα = [A∪(S
{Cβ :β < α})]τ θ,Cα satisfies (1), (3), (4) and, by Theorem 6, also (2). The setC=S
{Cα :α < m+} is τ θ-closed, in fact letx∈clτ θ(C), for everyV ∈ B(x) choose a point in clτ(V)∩C and let K be the set so obtained, clearly |K| ≤ m, therefore there exists an α < m+ such that K ⊂ Cα, then x ∈ clτ θ(K) ⊂ clτ θ(Cα) = Cα ⊂ C. Obviously |C| ≤ 2m, so to complete the proof it suffices to show that C = X. Let us suppose that y ∈ X −C, since X is functionally Hausdorff, then for anyx∈C there is a Ux ∈ B(x) such that y /∈ clτ(Ux); for every x ∈ X −C let Ux ∈ B(x) such that clτ(Ux)∩C = ∅ (C is τ θ-closed). {Ux}x∈X is an open cover of X, since wcd(X) ≤ m there is a B ⊂ X such that |B| ≤ m and X = S
{clτ(Ux) : x ∈ B}, clearly C ⊂ S{clτ(Ux) : x ∈ B∩C}. Since |B ∩C| ≤ m, there is an α < m+ such that B∩C⊂Cα. LetU ={Ux:x∈B∩C},U ⊂S{B(x) :x∈S{Cβ :β < α+ 1}},
|U| ≤m, y ∈X−S
{clτ(Ux) :Ux ∈ U} andCα+1−S
{clτ(Ux) :Ux ∈ U} =∅, a contradiction. HenceC=X and the proof is complete.
Remark 11. LetX be a functionally Hausdorff space and let wX be the com- pletely regular space which has the same points and continuous real valued func- tions as those of X. Clearly L(wX) ≤ wcd(X) for every functionally Haus- dorff space X. On the other hand, there exist functionally Hausdorff spaces X such that χ(X) < χ(wX) (see e.g. [9, Example 36]). I do not know if χ(wX)L(wX) ≤ χ(X)wcd(X) for every functionally Hausdorff space X; if this is the case, then Theorem 10 is a consequence of the Arkhangel’skii’s inequality quoted at the beginning.
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Dipartimento di Matematica Pura ed Applicata, Universit´a, 67100 L’Aquila, Italy (Received July 27, 1993)
