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The Non-Hausdorff Number of a Topological Space THE NON-HAUSDORFF NUMBER OF A TOPOLOGICAL SPACE IVAN S. GOTCHEV Abstract. We call a non-empty subset A of a topological space X finitely non-Hausdorff if for every non-empty finite subset F of A and every family fUx : x 2 F g of open neighborhoods Ux of x 2 F , \fUx : x 2 F g 6= ; and we define the non-Hausdorff number nh(X) of X as follows: nh(X) := 1 + supfjAj : A is a finitely non-Hausdorff subset of Xg. Using this new cardinal function we show that the following three inequalities are true for every topological space X d(X) (i) jXj ≤ 22 · nh(X); 2d(X) (ii) w(X) ≤ 2(2 ·nh(X)); (iii) jXj ≤ d(X)χ(X) · nh(X) and (iv) jXj ≤ nh(X)χ(X)L(X) is true for every T1-topological space X, where d(X) is the density, w(X) is the weight, χ(X) is the character and L(X) is the Lindel¨ofdegree of the space X. The first three inequalities extend to the class of all topological spaces d(X) Pospiˇsil'sinequalities that for every Hausdorff space X, jXj ≤ 22 , 2d(X) w(X) ≤ 22 and jXj ≤ d(X)χ(X). The fourth inequality generalizes 0 to the class of all T1-spaces Arhangel ski˘ı’s inequality that for every Hausdorff space X, jXj ≤ 2χ(X)L(X). It is still an open question if 0 Arhangel ski˘ı’sinequality is true for all T1-spaces. It follows from (iv) that the answer of this question is in the afirmative for all T1-spaces with nh(X) not greater than the cardianilty of the continuum. Examples are given to show that the upper bounds in (i) and (iii) are exact and that nh(X) cannot be omitted. 1. Introduction Let X be a topological space and for x 2 X let Nx denote the family of all open neighborhoods of x in X. For a nonempty subset A of X we denote by UA the set of all families U := fUa : a 2 A; Ua 2 Nag. A family Γ is centered if the intersection of any finitely many elements of Γ is non-empty. 2010 Mathematics Subject Classification. Primary 54A25, 54D10. Key words and phrases. Cardinal function, the non-Hausdorff number of a space, (max- imal) non-Hausdorff subset of a space, Arhangel0ski˘ı’sinequality, Pospiˇsil'sinequalities. This paper is respectfully dedicated to W. W. Comfort on the occasion of his 80th birthday. 1 2 IVAN S. GOTCHEV Recall that d(X) := minfjAj : A ⊂ X; A = Xg, w(X) := minfjBj : B is a base for Xg, L(X) := minfκ : every open cover of X has a subcover of cardinality ≤ κg, χ(x; X) := minfjVj : V is a local base for xg and χ(X) := supfχ(x; X): x 2 Xg. In 1937, Pospiˇsilproved (see [9]) that for every Hausdorff space X, (a) jXj ≤ 22d(X) ; d(X) (b) w(X) ≤ 222 ; and (c) jXj ≤ d(X)χ(X); and in 1969, Arhangel0ski˘ı (see [1]) answered a question of Alexandroff and Urysohn raised in 1923 by showing that for every Hausdorff space X, jXj ≤ 2χ(X)L(X). Since then many mathematicians have obtained similar inequalities for different classes of topological spaces but it is still unknown if 0 Arhangel ski˘ı’sinequality is true for all T1-topological spaces (see the survey paper [8]). In this paper we generalize Pospiˇsil'sinequalities for the class of all topo- 0 logical spaces and Arhangel ski˘ı’sinequality for the class of all T1-topological spaces and show that Arhangel0ski˘ı’sinequality is true for a very large class of T1-spaces. 2. The cardinal function nh(X) We begin with an example showing that Pospiˇsil'sinequality (c) is not always true for T1-spaces. Example 2.1. Let N denote the set of all positive integers and R be the set of all real numbers. Let S := f1=n : n 2 Ng and M := S[f0g be the subspace of R with the inherited topology. Then in M all points except 0 are isolated and limn!1 1=n = 0. Let α be an initial ordinal. We duplicate α many times the point 0 2 M i.e. we replace in M the point 0 with α many distinct points and obtain the set X := S [ α with topology such that for each β < α we have limn!1 1=n = β and the subspaces S and α with the inherited topology from X are discrete. Then the set f1=n : n 2 Ng is dense in X (hence d(X) = !), χ(X) = !, and if α > 2! then jXj > d(X)χ(X) = !! = 2!. To be able to generalize Pospiˇsil'sand Arhangel0ski˘ı’sinequalities we need to introduce some new concepts. Definition 2.2. We will call a nonempty subset A of a topological space X finitely non-Hausdorff if for every non-empty finite subset F of A and every U 2 UF , \U 6= ;. The set A will be called maximal finitely non-Hausdorff subset of X if A is a finitely non-Hausdorff subset of X and if B is a finitely non-Hausdorff subset of X such that A ⊂ B then A = B. We note that in a Hausdorff space X the only maximal finitely non- Hausdorff subsets of X are the singletons. Lemma 2.3. Every finitely non-Hausdorff subset of a topological space X is contained in a maximal finitely non-Hausdorff subset of X. THE NON-HAUSDORFF NUMBER OF A TOPOLOGICAL SPACE 3 Proof. It is a direct corollary of Zorn's lemma. Now we are ready to introduce the concept of a non-Hausdorff number of a topological space X. Definition 2.4. Let X be a topological space. We define the non-Hausdorff number nh(X) of X as follows: nh(X) := 1 + supfjAj : A is a (maximal) finitely non-Hausdorff subset of Xg. Remark 2.5. It follows from Definition 2.4 that X is a T2-space if and only if nh(X) = 2 and 2 < nh(X) ≤ 1 + jXj whenever X is a non-Hausdorff space. Also, if X is a topological space and A ⊂ X then nh(A) ≤ nh(X), and if X is an infinite set with topology generated by the open sets fX nfxg : x 2 Xg then X is a maximal finitely non-Hausdorff set and therefore nh(X) = jXj. Finally, using similar ideas as in Example 2.1 one can construct T1- spaces X with one or more of the following properties: (i) there exist maximal finitely non-Hausdorff subsets M and N of X such that jM \ Nj ≥ 0 and jMj, jNj, and jM \ Nj could have any cardinality which satisfy 0 ≤ jM \Nj ≤ jMj and 0 ≤ jM \Nj ≤ jNj; (ii) there exists a maximal finitely non-Hausdorff subset M and a point x 2 M such that M ( \fUx : Ux 2 Nxg and jMj = nh(X). We finish this section with two observations about finitely non-Hausdorff subsets of topological spaces. Lemma 2.6. Let X be a topological space and A be a finitely non-Hausdorff subset of X. Then A ⊂ \f\U : U 2 UF ; ; 6= F ⊂ A; jF j < !g. Proof. Let F be a nonempty subset of A, U0 2 UF , and G = \U0. Suppose that there exist a0 2 A such that a0 2= G. Then there is Wa0 2 Na0 such that Wa0 \ G = ;. Let Va0 = Wa0 if a0 2= F and Va0 = Ua0 \ Wa0 , where Ua0 2 U0 and Ua0 2 Na0 , if a0 2 F . Then the family U1 := fVa0 g [ fUa : Ua 2 U0; a 2 F n fa0gg has the property that \U1 = ; - a contradiction. Therefore A ⊂ \U for every U 2 UF and every nonempty subset F of A with jF j < !. Thus A ⊂ \f\U : U 2 UF ; ; 6= F ⊂ A; jF j < !g. Theorem 2.7. Let X be a topological space and A be a maximal finitely non- Hausdorff subset of X. Then A = \f\U : U 2 UF ; ; 6= F ⊂ A; jF j < !g. Proof. Let A be a maximal finitely non-Hausdorff subset of X. Then it follows from Lemma 2.6 that A ⊂ \f\U : U 2 UF ; ; 6= F ⊂ A; jF j < !g. Suppose that there is x0 2 \f\U : U 2 UF ; ; 6= F ⊂ A; jF j < !g n A. Then U \ (\U) 6= ; for every U 2 Nx0 , every U 2 UF and every nonempty finite subset F of A. Thus for the set A1 := A [ fx0g we have that if F ⊂ A1 with F 6= ; and jF j < !, and U 2 UF then \U 6= ;. Therefore, A1 is a finitely non-Hausdorff subset of X and A ( A1 - a contradiction with the maximality of A. 4 IVAN S. GOTCHEV 3. Some cardinal inequalities involving the non-Hausdorff number We begin with the generalization of Pospiˇsil'sinequalities (a), (b) and (c) for the class of all topological spaces. Theorem 3.1. Let X be a topological space. Then jXj ≤ 22d(X) · nh(X). Proof. Let D be a dense subset of X with d(X) = jDj and u = hn(X). For every nonempty finite subset A of X and every point x 2 A we choose a subset GA;x of D with x 2 GA;x in the following way: (i) If A is a finitely non-Hausdorff subset of X then GA;x := D; and (ii) If A is not a finitely non-Hausdorff subset of X then for each x 2 A we choose UA;x 2 Nx such that \fUA;x : x 2 Ag = ;.
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