Taras Banakh, Alex Ravsky Each regular paratopological group is completely regular Proceedings of the American Mathematical Society DOI: 10.1090/proc/13318 Accepted Manuscript This is a preliminary PDF of the author-produced manuscript that has been peer-reviewed and accepted for publication. It has not been copyedited, proof- read, or finalized by AMS Production staff. Once the accepted manuscript has been copyedited, proofread, and finalized by AMS Production staff, the article will be published in electronic form as a \Recently Published Article" before being placed in an issue. That electronically published article will become the Version of Record. This preliminary version is available to AMS members prior to publication of the Version of Record, and in limited cases it is also made accessible to everyone one year after the publication date of the Version of Record. The Version of Record is accessible to everyone five years after publication in an issue. EACH REGULAR PARATOPOLOGICAL GROUP IS COMPLETELY REGULAR TARAS BANAKH AND ALEX RAVSKY Abstract. We prove that a semiregular topological space X is completely regular if and only if its topology is generated by a normal quasi-uniformity. This characterization implies that each regular paratopological group is completely regular. This resolves an old problem in the theory of paratopo- logical groups, which stood open for about 60 years. Also we define a natural uniformity on each paratopological group and using this uniformity prove that each (first countable) Hausdorff paratopo- logical group is functionally Hausdorff (and submetrizable). This resolves another two known open problems in the theory of paratopological groups. According to a classical result of Pontryagin, for topological groups all separation axioms between T0 and T 1 are equivalent. For paratopological groups this equivalence does not hold anymore. We 3 2 recall that a paratopological group is a group G endowed with a topology making the group operation continuous. The additive group of real numbers R endowed with the topology τ0 = f;; Rg [ f(a; 1): a 2 Rg is an example of a paratopological group satisfying the separation axiom T0 but not T1. On the other hand, the topology τ1 = f;g [ fU ⊂ R : 9a 2 R (a; 1) ⊂ Ug turns the group R into a paratopological group satisfying the separation axiom T1 but not T2. Finally, the plane R×R endowed with the topology generated by the base B = f(a; b)g [ ((a; a + ") × (b; b + ")) : a; b 2 R; " > 0g is an example of a paratopological group which is Hausdorff but not regular (so, it satisfies the separation axiom T2 but not T3). These simple examples show that in contrast to topological groups, for paratopological groups the separation axioms T0;T1;T2;T3 are pairwise non-equivalent. For long time, the problem of the equivalence of the separation axioms T3 and T 1 (i.e., the 3 2 regularity and complete regularity) for paratopological groups remained unsolved (see [12, Question 1.2], [2, Problem 1.3.1], and [16, Problem 2.1]). Discussing this problem in the survey [16, x2], Tkachenko writes that it is \open for about 60 years" and \the authority of the question is unknown, even if all specialists in the area have it in mind". Six sections later in [16, x8] Tkachenko writes that \it is even more surprising that the following problem of Arhangel'skii [1, 3.11] is open: does every Hausdorff first countable paratopological group admit a weaker metrizable topology?" In this paper we will give surprisingly simple affirmative answers to both open problems. In fact, our principal results hold not only for paratopological groups but also for topological monoids with open shifts and, more generally, for topological spaces whose topology is generated by a normal quasi-uniformity. Now let us recall the necessary definitions related to quasi-uniformities (see [6], [8] and [9] for more information). For a subset A ⊂ X of a topological space by clX A and intX A we denote the closure and interior ◦ ◦ of A in X. The sets clX A, intX A and intX clX A will be also denoted by A, A , and A , respectively. Let X be a topological space. A subset U ⊂ X × X containing the diagonal ∆X = f(x; y) 2 X × X : x = yg is called an entourage on X. Given two entourages U; V ⊂ X × X let UV = (x; z) 2 X × X : 9y 2 X such that (x; y) 2 U and (y; z) 2 V be their composition and U −1 = f(y; x):(x; y) 2 Ug be the inverse entourage to U. 1991 Mathematics Subject Classification. 54D10; 54D15; 54E15; 22A30. Key words and phrases. Tychonoff space, regular space, completely regular space, semiregular space, Hausdorff space, functionally Hausdorff space, semi-Hausdorff space, separation axiom, quasi-uniformity, paratopological group, topological monoid. The first author has been partially financed by NCN grant DEC-2012/07/D/ST1/02087. 1 ThisAlgebra+NT+Comb+Logi is a pre-publication version of this article, which may differ from the final published version. CopyrightMay 15 restrictions 2016 12:48:11 may apply. EDT Version 2 - Submitted to PROC 2 TARAS BANAKH AND ALEX RAVSKY For an entourage U ⊂ X × X, point x 2 X and subset A ⊂ X, the set B(x; U) = fy 2 X :(x; y) 2 S Ug is called the U-ball centered at x, and the set B(A; U) = a2A B(a; U) is the U-neighborhood of ◦ A. It will be convenient to denote the sets B(A; U), B(A; U)◦ and B(A; U) by B(A; U), B◦(A; U) ◦ and B (A; U), respectively. A quasi-uniformity on a set X is a family U of entourages on X such that: • for any U; V 2 U there exists W 2 U such that W ⊂ U \ V ; • for any U 2 U there is V 2 U such that VV ⊂ U; • for every U 2 U, any entourage V ⊂ X × X containing U belongs to U. A quasi-uniformity U on X is called a uniformity if U = U −1, where U −1 = fU −1 : U 2 Ug. A subfamily B ⊂ U is called a base of a quasi-uniformity U if each entourage U 2 U contains some entourage B 2 B. Each quasi-uniformity U on a set X generates a topology τU on X consisting of the sets W ⊂ X such that for each point x 2 W there is an entourage U 2 U such that B(x; U) ⊂ W . Then for every point x 2 X the family fB(x; U)gU2U is a neighborhood base at x. It is clear that the topological space (X; τU ) satisfies the separation axiom T1 if and only if the quasi-uniformity U is T separated in the sense that U = ∆X . It is known (see [8] or [9]) that the topology of any topological space X is generated by a suitable quasi-uniformity. We shall say that a quasi-uniformity U on a topological space X is continuous if the topology τU generated by U is contained in the topology of X. ◦ Definition 1. A quasi-uniformity U on a topological space X is defined to be normal if A ⊂ B(A; U) for any subset A ⊂ X and any entourage U 2 U. A topological space X is called normally quasi- uniformizable if the topology of X is generated by a normal quasi-uniformity. Proposition 1. Each continuous uniformity U on a topological space X is normal. ◦ Proof. Given a subset A ⊂ X and an entourage U 2 U, we have to prove that A ⊂ B(A; U) . Choose any entourage V 2 U such that V −1V ⊂ U. It follows that for every point a 2 A the ball B(a; V ) meets the set A and hence a 2 B(A; V −1). Then B(a; V ) ⊂ B(B(A; V −1); V ) = B(A; V −1V ) ⊂ ◦ B(A; U) ⊂ B(A; U) and hence a 2 B(A; U) . In the proof of the following theorem we shall use the classical method of the proof of Urysohn's Lemma. Theorem 1. If U is a normal quasi-uniformity on a topological space X, then for every non-empty subset A ⊂ X and every entourage U 2 U there exists a continuous function f : X ! [0; 1] such that ◦ A ⊂ f −1(0) ⊂ f −1[0; 1) ⊂ B(A; U) . 1 ! Proof. Choose inductively a sequence of entourages (Un)n=0 2 U such that U0 ⊂ U and Un+1Un+1 ⊂ Un for every n 2 !. Here ! stands for the set of non-negative integer numbers and N = ! n f0g is the set of positive integers. k n Let Q2 = f 2n : k; n 2 N; 0 < k < 2 g be the set of binary fractions in the interval (0; 1). P1 rn Each number r 2 Q2 can be uniquely written as the sum r = n=1 2n for some binary sequence 1 N (rn)n=1 2 f0; 1g containing finitely many units. The sequence (rn)n2N will be called the binary Plr ri expansion of r. Since r > 0, the number lr = maxfn 2 N : rn = 1g is well-defined. So, r = i=1 2i . 1 0 For an entourage V 2 U we put V = V and V = ∆X . For every r 2 Q2 consider the entourage r U r = U r1 ··· U lr ; 1 lr which determines the closed neighborhood B(A; U r) of A. q We claim that for any rational numbers q < r in Q2 the neighborhood B(A; U ) is contained in ◦ r r the interior B (A; U ) of the neighborhood B(A; U ).