Connectedness of Ordered Rings of Fractions of C(X) with the m-topology

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1 Filomat 31:18 (2017), Published by Faculty o Sciences and Mathematics, Univesity o Niš, Sebia Available at: Connectedness o Odeed Rings o Factions o C(X) with the m-topology A.R. Salehi a a Depatment o Science, Petoleum Univesity o Technology, Ahvaz, Ian Abstact. An ode is pesented on the ings o actions S 1 C(X) o C(X), whee S is a multiplicatively closed subset o C(X), the ing o all continuous eal-valued unctions on a Tychono space X. Using this, a topology is deined on S 1 C(X) and o a amily o paticula multiplicatively closed subsets o C(X) namely m.c. z-subsets, it is shown that S 1 C(X) endowed with this topology is a Hausdo topological ing. Finally, the connectedness o S 1 C(X) via topological popeties o X is investigated. 1. Intoduction In this pape, the ing o all (bounded) eal-valued continuous unctions on a completely egula Hausdo space X, is denoted by C(X) (C (X)). The space X is called pseudocompact i C(X) = C (X). Fo evey C(X) the set Z( ) = { C(X) : (x) = 0} is said to be zeo-set o and it s complement which is denoted by coz, is called cozeo-set o. Moeove, an ideal I C(X) is said to be z-ideal i o evey I and C(X), the inclusion Z( ) Z( ) implies that I. u C(X) is a unit (i.e., u has multiplicative invese) i and only i Z(u) = and it is not had to see that an element o C(X) is zeo-diviso i and only i int X Z( ). The set o all units and the set o all zeo-divisos o C(X) ae denoted by U(X) and Zd(X) espectively. Let βx and υx be the Stone-Čech compactiication and the Hewitt ealcompactiication o the space X, espectively. Fo evey C (X) the unique extension o to a continuous unction in C(βX) is denoted by β and o each p βx, M p = { C(X) : p cl βx Z( )} (M p = { C (X) : β (p) = 0}) is a maximal ideal o C(X) (C (X)) and also, evey maximal ideal o C(X) (C (X)) is pecisely o the om M p (M p ), o some p βx. Moeove, o evey p βx, O p = { C(X) : p int βx cl βx Z( )} is the intesection o all pime ideals o C(X) which ae contained in M p. In act, we have; Lemma 1.1. ([7, Theoem 7.15]) Evey pime ideal P in C(X) contains O p o some unique p βx, and M p is the unique maximal ideal containing P. Wheneve p X, the ideals M p and O p will be the sets { C(X) : p Z( )} and { C(X) : p int X Z( )} espectively and in this case, they ae denoted by M p and O p. A maximal ideal M o C(X) is called eal wheneve the esidue class ield C(X) M is isomophic with the eal ield R. Thus, o evey p υx, Mp is a 2010 Mathematics Subject Classiication. Pimay 54C40; Seconday 54C35, 13B30 Keywods. Continuous unctions, connectedness, multiplicatively closed, m-topology Received: 04 Decembe 2016; Accepted: 06 Mach 2017 Communicated by Ljubiša Kočinac addess: a..salehi@put.ac.i (A.R. Salehi)

2 A.R. Salehi / Filomat 31:18 (2017), eal maximal ideal, and convesely evey eal maximal ideal o C(X) is pecisely o the om M p o some p υx. Moeove, M p C (X) = M p i and only i p υx, see 7.9 (c) in [7]. Let R be a commutative ing with unity and suppose that S is a multiplicatively closed subset o biely an m.c. subset o R. Hee S 1 R is the ing o all equivalence classes o the omal actions a s o a R and s S, whee the equivalence elation is the obvious one. Wheneve S is the set o all non-zeo-divisos o R, then S 1 R is called the classical ing o quotients o R. An m.c. subset T o R is called satuated wheneve a, b R and ab T imply that a and b belong to T. Fo an abitay m.c. subset S o R, the intesection o all satuated m.c. subsets o R which contain S, is called satuation o S and is denoted by S. Using 5.7 in [11] we have S = R\ P. P Spec(R) P S= Lemma 1.2. ([11, Execise 5.12(iv)]) Fo an abitay m.c. subset S o a commutative ing R with unity, two ings S 1 R and S 1 R ae isomophic. In sequel, o evey m.c. subset S o C(X), the ing o actions S 1 C(X) is oten abbeviated as S 1 C. 2. An Ode Relation on S 1 C The m-topology on C(X) is deined by taking the sets o the om B(, u) = { C(X) : (x) (x) < u(x), x X} as a base o the neighbohood system at, o each C(X), whee u uns though the set o all positive units o C(X). This topology on C(X) which is denoted by C m (X), was ist intoduced in [9] and studied moe in [1 3, 5, 8, 12]. To deine a topology on S 1 C, simila to the m-topology on C(X), we need an odeing to make S 1 C a lattice-odeed ing. We deine the ode elation on S 1 C as ollows: Deinition 2.1. Fo S 1 C, we deine 0 i thee exists t S such that 0 (t2 )(x) o all x X. Clealy 0 i and only i 0 ( )(x) o all x coz t, o some t S. This deinition is simila to the amilia deinition o ode on C(X). But hee we conside estiction o each on a cozeo-set o X instead o X itsel. To see that the ode is well deined, let, s S 1 C, = s and 0. Then thee exist p, q S such that q s = q and 0 p 2. Now, the inequality 0 (q 2 s 2 )(p 2 ) implies that 0 (p 2 sq)(q s) = (p 2 sq)(q ) = (p 2 2 q 2 )(s ) and since pq S, we conclude that 0 s. Poposition 2.2. let S be an m.c. subset o C(X), then (S 1 C, ) is a lattice-odeed ing. Poo. Clealy o evey S 1 C i 0 and 0, then = 0. Now, suppose that, s S 1 C, 0 and 0 s. Thee exist t 1, t 2 S such that 0 on coz t 1 and 0 s on coz t 2. Theeoe, 0 2 s 2 (s s ) and 0 2 s 2 ( s ) on coz t 1 t 2 and thus, 0 s2 + 2 s 2 s 2 S 1 C is lattice, it can be shown that = + s and 0 s 2 s 2 s = 2 s s 2 = s2 2 s 2 2 s 2 s 2 = s2 2 s s 2 2. = s on coz t 1t 2. To pove that

3 A.R. Salehi / Filomat 31:18 (2017), I S is an m.c. subset o a commutative ing R, then o evey n N, the set S n = {s n : s S} is an m.c. subset o R and clealy two ings (S n ) 1 R and S 1 R ae isomophic. In act, the map i n ( ) = n 1 is an n isomophism om S 1 R onto (S n ) 1 R. Now we deine an odeing on (S 2 ) 1 C as ollows; Deinition 2.3. Fo evey (S2 ) 1 C, we deine 0 i thee exists t S2 such that 0 t(x) (x) o all x X. I S is an m.c. subset o C(X) then S 2 { S : 0 }. Theeoe 0 i and only i 0 on coz t o some t S. Simila to Deinition 2.1, it can be shown that ((S 2 ) 1 C, ) is a lattice-odeed ing. Moeove, we have the ollowing esult whose poo is let to the eades. Poposition 2.4. Let S be an m.c. subset o C(X). Two ings (S 1 C, ) and ((S 2 ) 1 C, ) ae lattice isomophic. In act, the map i 2 ( ) = om S 1 C onto (S 2 ) 1 C is an isomophism and also ode-peseving, i.e., 2 s i and only i s. 2 s 2 Now using the above poposition, without loss o geneality, o evey lattice-odeed ing (S 1 C, ) we can assume that each membe o S is non-negative. In addition, we can conside 0 wheneve 0 on coz t o some t S. Deinition 2.5. A subset S o C(X) is called z-subset wheneve, C(X) and S, then Z( ) = Z( ) implies that S. Example 2.6. The set C(X)\Zd(X) = { C(X) : int X Z( ) = } o all non-zeo-diviso elements o C(X), is a multiplicatively closed z-subset (o biely an m.c. z-subset) o C(X). Anothe example o m.c. z-subset is U(X) = { C(X) : Z( ) = }, the set o all units o C(X). I {P λ } λ Λ is a amily o pime z-ideals o C(X), then S = C(X)\ λ Λ P λ is also an m.c. z-subset o C(X). Note that wheneve P is a pime ideal o C(X) which is not z-ideal, then S = C(X)\P is a satuated m.c. subset o C(X) which is not a z-subset. Poposition 2.7. I S is an m.c. z-subset o C(X), then the set T := { C(X) : Z( ) Z(s) satuation o S. o some s S} is the Poo. We show that T is the smallest satuated m.c. subset containing S. Fist, note that T is a satuated m.c. subset o C(X) containing S. In act, i, T then thee exist s 1, s 2 in S such that Z( ) Z(s 1 ) and Z( ) Z(s 2 ). Theeoe, Z( ) = Z( ) Z( ) Z(s 1 s 2 ) which implies T. Moeove, i T then Z( ) Z(s), o some s S. Thus Z( ) Z(s) and also Z( ) Z(s) which imply that, T. Next, let T be a satuated m.c. subset o C(X) containing S and suppose that T. Hence Z( ) Z(s), o some s S and thus Z( s) = Z( ) Z(s) = Z(s). Since S is a z-subset, s S T and so T, i.e., T T which complete the poo. Coollay 2.8. Let S be an m.c. subset o C(X). S is a satuated m.c. z-subset i and only i o evey C(X) and s S, the inclusion Z( ) Z(s) implies that S. Coollay 2.9. The satuation o evey m.c. z-subset o C(X) is a z-subset. Example Let (x) = x 1 be a unction o C(R). Then S 1 = {1,, 2,...} is an m.c. subset o X which is not z-subset no satuated. In act, S 2 = { C(R) : Z( ) = o Z( ) = {1, 1}} is the smallest m.c. z-subset o C(R) containing S 1 and o satuation o S 2 we have Moeove, it is easy to see that S 1 S 2 S 2. S 2 = { C(R) : Z( ) {1, 1}}.

4 A.R. Salehi / Filomat 31:18 (2017), Similaly to the ode elation, o evey S 1 C we deine 0 < i 0 < on coz t o some t S. Poposition The set U + = { S 1 C : 0 < } is closed with espect to the opeations and. Moeove, i S is an m.c. z-subset, then evey membe o U + is a unit o S 1 C. Poo. I, s U +, then thee exist t 1, t 2 S such that 0 < on coz t 1 and 0 < on coz t 2. Since 0, s we have 0 < s on coz t 1 t 2 s which implies that 0 < s s = s. To pove the second pat o the poposition, let 0 <. We have 0 < on coz t o some t S and so coz t coz. Theeoe, coz t = coz t and since S is an m.c. z-subset, then t S. Now, = t t S 1 C implies that is a unit. 3. The m-topology on S 1 C Beoe deining the m-topology on S 1 C, we note that = ( ) = ( ) =. Now, o each S 1 C and each u t U + i we conside the set B(, u t ) := { s : s < u t }, then clealy we have: B(, u t ) = { s : (x) s (x) < u (x) o all x coz q coz stu o some q S}. t The collection B = {B(, u t ) : S 1 C and u t U + } is a base o a topology on S 1 C. In act, B(, u t ) and B(, u t v s ) B(, u t ) B(, v s ) o evey u t, v s U+. Moeove, i s B(, u t ), then p q := u t s U+ and we have B( s, p q ) B(, u t ). As the m-topology on C(X), this topology on S 1 C is called the m-topology and S 1 C endowed with this topology is denoted by S 1 m C. This topology is in act a genealization o the m-topology on C(X). Note that wheneve S = U(X) then S 1 m C = C m (X). Recall that a topological ing is simply a ing unished with a topology o which its algebaic opeations ae continuous, see [13]. We also notice that a Hausdo topological ing is completely egula, see in [6]. To pove that S 1 m C is a Hausdo topological ing we need the ollowing lemmas. Lemma 3.1. Let S be an m.c. z-subset o C(X). Fo evey 0 S 1 C thee exists s S 1 C such that 0, s 1 and = s. Poo. Conside s = 1++ and = 1++. Clealy Z(s) = Z() implies s S and we have s = = =. Lemma 3.2. I S is an m.c. z-subset o C(X), then the set {B(, v 1 ) : C(X),, v S and 0 v 1} is a base o the m-topology on S 1 C. Poo. By Lemma 3.1, o each B(, u t ) thee exist v, s S such that 0 v, s 1, and u t = v s. But s(x)v(x) v(x) o all x coz sv, then v 1 v s and so B(, v 1 ) B(, v s ) = B(, u t ). Poposition 3.3. Let S be an m.c. z-subset o C(X). Then S 1 m C is a Hausdo topological ing. Poo. To pove the continuity o addition and multiplication, let, s S 1 C and u 1 U+. Then ( ( u ) ( + B, 2 u )) ( B 1 s, 2 B 1 + s, u ) 1. ( B (, v ) ( B 1 s, v ) ) ( B 1 s, u ) 1

5 A.R. Salehi / Filomat 31:18 (2017), whee v 1 U+ such that ( u s ) v 1 < u 1. In act, i we conside w := ( u s ), then 1 1 < w U+, w 1 < 1 1 and w 1 U +. Now, it is enough to take v 1 = w 1 u 2. To show that S 1 m C is Hausdo, let, s S 1 C and s. Thus, s on coz s and so coz s coz ( s ). Theeoe, coz s = coz s( s ) and since S is an m.c. z-subset and s S, we have t := s( s ) S. Now, it is not had to see that B(, t ) and B( 2 2 s 2 s, t ) ae disjoint. 2 2 s 2 Coollay 3.4. Let S be an m.c. z-subset o C(X). Then S 1 C with the m-topology is a completely egula Hausdo space. 4. Connectedness o S 1 m C In this section, in imitate o [2], we ist ind the component o zeo in S 1 m C, whee S is an m.c. z-subset. Next using this, we give a necessay and suicient condition o connectedness o S 1 m C. Deinition 4.1. A membe S 1 C is called bounded i thee exists k N such that k 1, i.e., (x) k (x) o all x coz t o some t S. Clealy the set (S 1 C) o all bounded elements o S 1 C is a subing o S 1 C. Lemma 4.2. (S 1 C) is a clopen subset o S 1 m C. Poo. I (S 1 C), then B(, 1 1 ) (S 1 C). In act, s < 1 1 implies that s < k k N and hence s is bounded. On the othe hand, i (S 1 C), then B(, 1 1 ) (S 1 C) =. o some Lemma 4.3. J ψ = { S 1 C : s t is bounded o each s t U+ } is an ideal o S 1 C. Poo. It is not had to see that J ψ is closed with espect to addition. Let J ψ, s S 1 C and p q U+. We claim that p sq 0 < (1 + )p on coz t. Theeoe, (1+ )p that p sq = ( (1+ )p sq is bounded and so s J ψ. Since p q U+, 0 < p on coz t o some t S and so sq U +. Now by ou hypothesis, (1+ )p sq is bounded which implies ) ( ) p 1+ is bounded and sq is bounded as well. Using Lemmas 3.1 and 3.2 we have J ψ = { S 1 C : t is bounded, t U+, 0 t 1} Lemma 4.4. Let S be an m.c. z-subset o C(X) and conside ϕ (a) = a is continuous. J ψ. The unction ϕ : R S 1 C deined by Poo. Using Lemma 3.2, o evey a R and v 1 U+, we must show that ϕ 1 (B( a )) contains a neighbohood o a in R. Since J ψ, thee exists k N such that (a 1 k, a + 1 k ) is contained in ϕ 1(B( a 1, v 1 )). In act, b (a 1 k, a + 1 k b a < v 1, i.e. b ϕ 1(B( a, v 1 )). 1 v k 1 The ollowing theoem is in act a genealization o Coollay 3.3 in [2]., v 1. Now, we show that the inteval ) implies that b a 1 1 v 1 k k 1 = 1 1 Theoem 4.5. Let S be an m.c. z-subset o C(X). The ideal J ψ is the component o zeo in S 1 m C. and hence

6 A.R. Salehi / Filomat 31:18 (2017), Poo. Fist, since R is connected, using Lemma 4.4, ϕ (R) is a connected set containing 0 o evey J ψ. Theeoe, J ψ = ϕ J (R) is a connected set containing 0. Next, I I is the component o 0 in S 1 m C, ψ then J ψ I. Moeove, since S 1 m C is topological ing, I is an ideal o S 1 m C. To complete the poo, it is enough to show that I J ψ. On the contay, let I\J ψ. By Lemma 4.3, thee exists s t U + such that s t (S 1 C). Conside the sets I (S 1 C) and I\(S 1 C). By Lemma 4.2, these two sets ae open in I and since 0 I (S 1 C) and s t I\(S 1 C), they ae non-empty disjoint open subsets o the connected set I, a contadiction. Coollay 4.6. Let S be an m.c. z-subset o C(X). S 1 m C is connected i and only i S 1 m C = J ψ, i.e., o evey C(X) and each S, thee exist k N and t S such that (x) k(x) o all x coz t. Motivated by the pevious coollay, we ae going to investigate the connectedness o S 1 m C via topological popeties o X o some paticula m.c. z-subsets o X. Fo example, let p βx and put S p = C(X)\M p o moe geneally, suppose that A βx and S A := C(X)\ p A M p. Clealy S A is an m.c. z-subset o C(X) and S A = { C(X) : p cl βx Z( ) o each p A} = { C(X) : A cl βx Z( ) = }. Now, it is natual to ask the ollowing questions. When is the topological ing (S A ) 1 m C connected? what can we say about the connectedness o (S A ) 1 m C i we eplace p A M p in S A by an abitay union o amily o paticula pime ideals o C(X)? We will addess such questions in the next section. 5. Connectedness o S 1 C with the m-topology A In this section, we study the connectedness o S 1 m C, whee S = C(X)\ λ Λ P λ, and {P λ } λ Λ is a amily o pime z-ideals o C(X). Using this, we conclude that C(X) with the m-topology is connected i and only i X is pseudocompact. Also, It is shown that the classical ing o quotients o C(X) endowed with the m-topology, is connected i and only i evey dense cozeo-set o C(X) is pseudocompact. We use the ollowing lemma equently. But, beoe that, we eview some esults which ae needed in sequel. Fist, notice that o evey C(X) we have coz βx\cl βx Z( ) cl βx coz. (1) The poo o the ist inclusion is clea. To pove the second, let x cl βx Z( ). Thee exists an open neighbohood G o x in βx such that G βx\z( ). Now, o an abitay open subset H o βx containing x, we have X (G H) (X H) ( βx\z( ) ) = H coz which implies that x cl βx coz. Next, by pat (1), we conclude that cl βx coz = cl βx (βx\cl βx Z( )) = βx\int βx cl βx Z( ). (2) Finally, i C (X), then coz = X coz β and in this case, we have Using pat (2), the next lemma is now evident. cl βx coz = cl βx coz β = βx\int βx Z( β ). Lemma 5.1. Let C(X) and p βx. O p i and only i p cl βx coz. Poposition 5.2. Let S be an m.c. z-subset o then S 1 m C is disconnected. C(X) and conside S p := C(X)\M p, o some p βx \ υx. I S p S

7 A.R. Salehi / Filomat 31:18 (2017), Poo. Let p βx\υx. By 8.7.(b) in [7], thee exists C (X) such that Z() =, while β (p) = 0. Since S is a z-subset and Z() = Z(1), then S and so 1 S 1 C. To complete the poo, we claim that 1 is unbounded on coz t o evey t S. Let S be the satuation o S. Recall that S = C(X)\ P S= P whee each P is a pime ideal o C(X). Futhemoe, o evey pime ideal P which doesn t intesect S, we have P C(X)\S C(X)\S p = M p. Now, o evey t S, t S S and so, thee exists a pime ideal P M p such that t P and consequently t O p. Thus by Lemma 5.1, p cl βx coz t and hence thee exists a net {x λ } contained in coz t = coz t coz which conveges to p. Since β is continuous, β (x λ ) β (p) = 0 and this implies that the unction conveges to zeo on coz t coz and so the action 1 S 1 C is not bounded on coz t coz. Theeoe, the claim is tue and so S 1 m C is disconnected. Poposition 5.3. Let P be a pime z-ideal o C(X) and suppose that S = C(X)\P. The topological ing S 1 m C is connected i and only i P is a eal maximal ideal. Poo. We ist pove the necessity. By contay, assume that P is not eal maximal ideal. Now, using 7.15 in [7], let p βx and M p be the unique maximal ideal o C(X) containing P. We conside two cases: Case 1. p βx\υx. In this case, Poposition 5.2 implies that S 1 m C is disconnected, a contadiction. Case 2. let p υx. In this case, using 7.9.(c) in [7], we have M p C (X) = M p. On the othe hand, P M p by ou assumption. Then thee exists a unction C (X) such that M p \P and so 1 S 1 C. Moeove, M p C (X) implies β (p) = 0. Now, o evey t S, t P which shows that t O p and hence p cl βx coz t, by Lemma 5.1. Finally, simila to the poo o the Poposition 5.2, we conclude that 1 is unbounded on coz t coz t and consequently using the Coollay 4.6, S 1 m C is not connected, a contadiction. Next, to pove the suiciency, let p βx, M p be a eal maximal ideal o C(X) and S = C(X)\M p. suppose that S 1 C. By Lemma 3.1, we can assume that, C (X). Since M p and M p is eal, we have M P, by 7.9.(c) in [7], and hence β (p) 0. Moeove, o evey C(X), β (p) does not appoach to ininity. Now, conside the open subset H = {x coz β : β (x) β (p) < 1} o coz β βx. We obseve that H is an open β β neighbohood o p in βx and since {coz t β : t C (X)} is a base o the space βx, thee exists t C (X) such that p coz t β H. Thus, o evey x coz t β, β (x) < β (p) + 1 and hence o evey x X coz t β = coz t, β β we have (x) < β (p) + 1 which implies that β is bounded on coz t. Theeoe, by Coollay 4.6, S 1 C with the m-topology, i.e., S 1 m C is connected. The ollowing esult is an immediate consequence o the pevious poposition. Coollay 5.4. Let p βx. S 1 p C with the m-topology is connected i and only i p υx. By 8A.4 in [7], υx = βx i and only i X is pseudocompact. Using this and coollay 5.4 the ollowing esult is now evident. Coollay 5.5. S 1 p C with the m-topology is connected o evey p βx i and only i X is pseudocompact. Recall that wheneve S is the satuation o an m.c. subset S o C(X), then two ings S 1 C and ( S) 1 C ae isomophic. By Coollay 2.9, the satuation o evey m.c. z-subset S o C(X) is a z-subset. I we conside S = C(X)\ λ Λ P λ whee {P λ } λ Λ is a amily o pime ideals o C(X), then o evey λ Λ, we have P λ S = and convesely, o each pime ideal P disjoint om S thee exists λ Λ such that P = P λ. Deinition 5.6. An ideal I o C(X) is called eal wheneve evey maximal ideal containing I, is eal. As 7O in [7], o an ideal I in C(X) i we deine θ(i) = {p βx : I M P }, then θ(i) = I cl βx Z( ). Thus, an ideal o C(X) is eal ideal i and only i θ(i) υx o equivalently I cl βx Z( ) υx. Poposition 5.7. Let {P λ } λ Λ be a amily o pime z-ideals o C(X) and take S := C(X)\ λ Λ P λ. Then S is an m.c. z- subset o C(X) and i S 1 m C is connected, then o evey λ Λ the ideal P λ is eal. Moeove, λ Λ P λ = p A M p whee A = λ Λ θ(p λ ).

8 A.R. Salehi / Filomat 31:18 (2017), Poo. By contay, suppose that S 1 m C is connected but at least one o the pime ideals is not eal. Thus, thee exists λ Λ and p βx\υx such that P λ M P. Since p υx, thee is a unction C (X) such that Z() = and β (p) = 0. Now, simila to the poo o Poposition 5.2, we conclude that 1 S 1 C and o evey t S we have t O p, since t P λ. So by Lemma 5.1, p cl βx coz t. Theeoe, 1 is not bounded on coz t and thus S 1 m C is not connected, by Coollay 4.6, a contadiction. To pove the last pat o the poposition, by contay, let p A M p \ λ Λ P λ. As above, o evey t S it can be shown that 1 is unbounded on coz t and so S 1 m C is disconnected, a contadiction. Coollay 5.8. Let p βx and {P p λ } λ Λ be a amily o pime z-ideals o C(X) contained in the maximal ideal M p and suppose that S = C(X)\ λ Λ P p. Then S 1 λ m C is connected i and only i p υx and M p = λ Λ P p λ. Coollay 5.9. Let A βx and suppose that S A = C(X)\ p A M p. I S 1 C with the m-topology is connected, then A υx. The ollowing theoem which is in act a genealization o Coollay 5.4, shows that wheneve A is a compact subset o βx, the convese o the pevious coollay is also tue. But, we wee unable to answe the convese o the coollay. Theoem Let A be a compact subset o βx and conside S A = C(X)\ p A M p. Then S 1 C with the m-topology A is connected i and only i A υx. Poo. Necessity is clea by Coollay 5.9. To pove the suiciency, let A be a compact subset o υx. Using Coollay 4.6, it is enough to show that o evey S 1 A C, thee exists t S A such that is bounded on coz t. Since S A, then o evey p A υx, M p C (X) = M p and so β (p) 0. Moeove, p υx implies that β (p) and thus o each p A, β (p) is a eal numbe. As in the poo o Poposition 5.3, β the subset H = {x coz β : β (x) β (p) < 1} is an open neighbohood o p in coz β and hence in βx as β β well. Thus, thee exists t C (X) such that p coz t β p H coz β and so we conclude that is bounded on coz t p. In act, o evey x coz t p we have (x) < β (p) + 1. Now, since A is compact and A β p A coz t β p, thee ae unctions t p1,..., t pn in C (X) such that A n i=1 coz tβ p i. We claim that t = t 2 p t 2 p n is the unction which we look o. Fist, note that o evey p A we have t M p. Othewise, i o some q A we have t M q, then Z(t) Z(t pi ) (1 i n) implies that t pi M q o evey 1 i n, (since M q is a z-ideal) which contadicts q A n i=1 coz tβ p i. Next, because is bounded on evey coz t p i (1 i n), it is bounded on coz t = coz (t 2 p t 2 p n ) = n i=1 coz t p i too, which completes the poo. Wheneve a subset A o X is completely sepaated om evey zeo-set disjoint om it, in paticula, i A is a zeo-set o a C-embedded subset o X, then o evey C(X), cl βx A cl βx Z( ) = i and only i A cl βx Z( ) =, see Theoems 1.18 and 6.5 in [7]. Theeoe, S A = S clβx A and since cl βx A is a compact subset o βx, the ollowing esult is now evident by Theoem Coollay Let a subset A X be completely sepaated om evey zeo-set disjoint om it. Then S 1 C with the A m-topology is connected i and only i cl βx A υx. I we conside S = C(X)\ p βx M p, then S is the set o all units o C(X) and so S 1 m C = C m (X). Theeoe, by Theoem 5.10, C m (X) is connected i and only i βx υx. Now, using 8A.4 in [7], the ollowing esults is evident. Coollay ([2, Poposition 3.12]) C(X) with the m-topology is connected i and only i X is pseudocompact.

9 A.R. Salehi / Filomat 31:18 (2017), Using Poposition 5.7 and Theoem 5.10, we conclude the pape by anothe poo o Coollay 3.11 in [3]. Fist, we ecall that a point p X is called an almost P-point, i evey G δ -set (zeo-set) containing p has nonempty inteio and a space X is called an almost P-space i each point o X is an almost P-point. Thus, X is an almost P-space i and only i evey non-zeo-diviso o C(X) is unit, i.e., U(X) = C(X)\Zd(X) = C(X)\ P, whee P is a pime ideal o C(X) contained in Zd(X). It is poved that p X is an almost P-point i and only i wheneve C(X) and p Z( ) imply that p cl X int X Z( ). In act, i p is an almost P-point, then M p Zd(X) and thus, o evey C(X) i p Z( ), then the ideal (O p, ) geneated by O p { }, is contained in M p. Now, using Coollay 3.3 in [4] we conclude that p cl X int X Z( ). See [10] o moe inomation about almost P-spaces. Coollay ([3, Coollay 3.11]) The classical ing o quotients o C(X) with the m-topology is connected i and only i X is a pseudocompact almost P-space. Poo. Let S 1 C be the classical ing o quotients o C(X) and o evey p βx, suppose that {P p λ } λ Λ p is the amily o all pime ideals o C(X) contained in M p Zd(X). It is not had to see that Zd(X) = λ Λ p P p λ p βx and so, S = C(X)\ λ Λ p P p. Now, Using Poposition 5.7, i S 1 λ m C is connected, then evey P p λ p βx is eal ideal which implies that βx υx, i.e., X is pseudocompact. On the othe hand, since o evey λ Λ p we have θ(p p λ ) = {p}, using the same poposition, we conclude that Zd(X) = p βx M p. Thus, each non-unit o C(X) is zeo-diviso and this means that X is almost p-space. Convesely, let X be pseudocompact almost P-space. Since X is an almost P-space, Zd(X) = p βx M p and by pseudocompactness o X we conclude that βx = υx. Now, by Theoem 5.10, S 1 m C is connected o S = C(X)\Zd(X). Acknowledgement. The autho is gateul to the eeee o cetain comments and coections towads the impovement o the pape. Reeences [1] F. Azapanah, F. Manshoo, R. Mohamadian, A genealization o the m-topology on C(X) ine than the m-topology, Filomat, to appea. [2] F. Azapanah, F. Manshoo, R. Mohamadian, Connectedness and compactness in C(X) with the m-topology and genealized m-topology, Topology Appl. 159 (2012) [3] F. Azapanah, M. Paimann, A.R. Salehi, Connectedness o some ings o quotients o C(X) with the m-topology, Comment. Math. Univ. Caolinae 56 (2015) [4] F. Azapanah, A. R. Salehi, Ideal stuctue o the classical ing o quotients o C(X), Topology Appl. 209 (2016) [5] G. Di Maio, L. Hola, D. Holy, R.A. McCoy, Topology on the space o continuous unctions, Topology Appl. 86 (1998) [6] R. Engelking, Geneal Topology, Sigma Se. Pue Math., vol. 6, Heldemann Velag, Belin, [7] L. Gillman, M. Jeison, Rings o Continuous Functions, Spinge-Velag, New Yok, [8] J. Gomez-Peez, W.W. McGoven, The m-topology on C m (X) evisited, Topology Appl. 153 (2006) [9] E. Hewitt, Rings o eal-valued continuous unctions I, Tans. Ame. Math. Soc. 48(64) (1948) [10] R. Levy, Almost P-spaces, Cannad. J. Math. 29 (1977) [11] R.Y. Shap, Steps in Commutative Algeba, Cambidge Univesity Pess, [12] E. van Douwen, Nonnomality o heeditay paacompactness o some spaces o eal unctions, Topology Appl. 39 (1991) [13] S. Wane, Topological Rings, Noth Holland Math. Stud., 1993.