PUBLICATIONS DE L’INSTITUT MATHÉMATIQUE Nouvelle série, tome 99(113) (2016), 217–225 DOI: 10.2298/PIM1613217B

MAXIMAL TYCHONOFF SPACES AND NORMAL ISOLATOR COVERS

C. K. Basu and S. S. Mandal

Abstract. We introduce a new kind of cover called a normal isolator cover to characterize maximal Tychonoff spaces. Such a study is used to provide an alternative proof of an interesting result of Feng and Garcia-Ferreira in 1999 that every maximal Tychonoff space is extremally disconnected. Maximal tychonoffness of subspaces is also discussed.

1. Introduction

In the poset A(X), of all topologies on a given set X, having the property P, a topological space (X, τ) is maximal P provided that τ is a maximal element in A(X). In [6], it had been shown that a topological space (X, τ) is maximal P if and only if every continuous bijection from a space (Y, τ_{1}) with the property P to (X, τ) is a homeomorphism. In 1943 Hewitt [15] and in 1947 Vaidyanathaswamy [29] had independently proved that every compact Hausdorff space is maximal compact. Vaidyanathaswamy [29] put forward a question if there exists any non-Hausdorff maximal compact space. One year later in 1948 Hing tong [28] answered affirmatively Vaidyanathaswamy’s question. In the same year Ramanathan [21] characterized maximal compact spaces as those whose compact subsets are precisely the closed sets. Levine [17] answered affirmatively the question of Vaidyanathaswamy by establishing that a one point compactification of rationals with the usual topology is a non-Hausdorff maximal compact space. In the same paper he exhibited that maximal compact topologies are not productive. On the other hand, Mioduszewski and Rudolf [18] demonstrated necessary and sufficient conditions for an absolutely closed (or H-closed) space to become maximal absolutely closed.

Thron [27] and Aull [1] investigated maximal countably compact spaces. Aull, in fact, strengthened the result: a first countable Hausdorff countably compact space is maximal countably compact and minimal first countable Hausdorff of Thron [27].

2010 Mathematics Subject Classification: 54D05, 54E15, 54E99.

Key words and phrases: star refinement, refinement, normally open cover, normal isolator cover, stronger topology, maximal uniformizable, extremally disconnected.

Communicated by Miloš Kurilić.

In 1971, Cameron [6] and in 1973 Raha [20] investigated exhaustively the various aspects of certain maximal P spaces, where P = Lindeloff, countably compact, sequential compact, pseudocompact, lightly compact or connected. Thomas [26] had also discussed maximal connected topologies. Cameron characterized maximal QHC spaces [7] and maximal pseudocompact spaces [5, 9] and in [8] he shown that the maximal topologies of a class of topologies which include lightly compact and QHC spaces are submaximal and T_{1} spaces. In 1999, Kennedy and McCartan [16] investigated spaces which are maximal with respect to a semiregular property and showed new characterizations of maximal QHC spaces and maximal pseudocompact spaces. In 1997, Guthrie, Stone and Wage [13] investigated topologies, which are maximal connected Hausdorff and in 1998, Shakhmatov, Tkacenko, Tkachuk, Watson and Wilson [23] showed that neither first countable nor Cech-complete spaces are maximal Tychonoff connected also in 2007, Zelenyuk [31] investigated almost maximal spaces. In addition, interesting behaviors of some of Maximal topologies and their applications are found in the papers [10, 11, 14, 19, 22, 24, 25].

Considering the usefulness and importance of uniformizability (=Tychonoffness in Hausdorff spaces) and the above observations about maximal topologies of various kinds of topological properties, this article is devoted to study maximal uniformizable(=Maximal Tychonoff) spaces. Several characterizations of such spaces have been given in terms of refinement of normally open covers as well as newly introduced normal isolator covers. As a consequence, we provide an alternative proof of the already existing interesting result of Feng, Garcia-Ferreira [12] that every maximal Tychonoff space is extremally disconnected. Maximal uniformizability with respect to subspaces has also been discussed.

2. Preliminaries

The symbol X or (X, τ) denotes a topological space without any isolated points which is T_{2} and the base set X is infinite, unless explicitly stated. For two covers U and V of X, U is called a refinement of V denoted by U < V if for each U ∈ U, there exists a V ∈ V such that U ⊂ V and we call U star refines V or U is a star refinement of V, denoted by U <^{⋆} V, if for each U ∈ U, there exists a V ∈ V such that St(U; U) ⊂ V, where St(U; U) = ⋃ {W ∈ U : W ∩ U ≠ ∅}. When U = {x}, we denote St(U; U) as St(x; U). We note that if U <^{⋆} V, then U < V.

A normal sequence of covers of X is a sequence of covers U_{1}, U_{2}, … of X such that U_{n+1} <^{⋆} U_{n}, for n = 1, 2, …; and a normal cover is a cover which is U_{1} in some normal sequence of covers [30, §36.9. p. 247]. An open cover U of a topological space X is normally open if and only if U = U_{1} in some normal sequence U_{1}, U_{2}, … consisting of open covers of X [30, §36.14. p. 248].

A collection μ′ of covers of a space X is a base for some covering uniformity on X if and only if it satisfies the condition that for U_{1}, U_{2} ∈ μ′ there is a U_{3} ∈ μ′ such that U_{3} <^{⋆} U_{1} and U_{3} <^{⋆} U_{2} [30, §36.3, p. 245]. It is well known that if μ′is a base for a covering uniformity μ on X, then {St(x; U) : U ∈ μ′} is a local base at x ∈ X in the uniform topology [30, §36.6, p. 246]. Also if X is any uniformizable topological space, then there is a finest uniformity on X, compatible with the topology of X, called the fine uniformity on X, denoted by μ_{F}, which has a base of all normally open covers of X. So a uniformizable space (X, τ) has at least one normally open cover consisting of proper subsets of X.

Lemma 2.1. If U_{1}, U_{2}, U_{3} are three covers of X such that U_{1} <^{⋆} U_{2} <^{⋆} U_{3}, then

U_{1} <^{⋆} U_{3}

Proof. The proof is obvious. □

Lemma 2.2. If {U_{1}, U_{2}, U_{3}, …} is a normal sequence of covers and if U_{k}, U_{m} ∈ {U_{1}, U_{2}, U_{3}, …}, then for a positive integer t greater than both of k and m, U_{t} <^{⋆} U_{k}

and U_{t} <^{⋆} U_{m}.

Proof. The proof is obvious. □

Theorem 2.1 (Hausdorff criterion [30]). For each x ∈ X, let B_{x}^{1} be a neighborhood base at x for the topology τ_{1} on X and B_{x}^{2} be a neighborhood base at x for the topology τ_{2} on X. Then τ_{1} ⊂ τ_{2} if and only if for each x ∈ X and each B^{1} ∈ B_{x}^{1}, there is some B^{2} ∈ B_{x}^{2} such that B^{2} ⊂ B^{1} [30, §4.8. p. 35].

Definition 2.1. For two covers U and V of X, we denote the intersection of U and V as U ∧ V and define it as U ∧ V = {U ∩ V : U ∈ U, V ∈ V} [30, §36.3. p. 245].

3. Maximal Tychonoff spaces

Recently, when a space (uniformizable or not) possessing a nontrivial proper uniformizable subtopology is investigated in [3] by Basu and Mandal, by the help of normal sequence of covers and star refinement of covers. A useful consequence of that investigation reflects that a sort of converse of A. H. Stone’s famous theorem is true when Basu and Mandal [3] established that a paracompact T_{2} space (X, τ) is either metrizable or (X, τ) has a nontrivial proper uniformizable subtopology, which is pseudometrizable. In the course of that study, disconnectedness is seen to play a major role, especially when that is of very strong in nature viz. zero-dimensionality, it is shown there that for a paracompact T_{2} space (X, τ) containing no isolated points, the cardinality of such nontrivial proper uniformizable subtopologies of (X, τ) is at least ℵ_{0}. In another paper [4], Basu and Mandal characterized minimal Uniformizable spaces in terms of normal sequence of covers and have shown that a minimal uniformizable non-indiscrete space is pseudometrizable. In this section, we investigate maximal uniformizable (=maximal Tychonoff ) spaces in terms of a new kind of cover called a normal isolator cover.

Definition 3.1. A cover U of (X, τ) is called an isolator cover of (X, τ) if St(x; U ∧ V) is infinite for every normally open cover V of (X, τ) and U is called a normal isolator cover if it is the first term of a normal sequence of isolator covers.

Clearly every open cover of (X, τ) is an isolator cover of (X, τ) and every normally open cover of (X, τ) is also obviously a normal isolator cover of (X, τ).

We further note that if τ_{1} is a topology on X such that τ_{1} ⊃ τ and if U is a normal isolator cover of (X, τ_{1}), then U is a normal isolator cover of (X, τ).

Definition 3.2. A Tyconoff (or uniformizable) space (X, τ) is called maximal Tyconoff [11] (or maximal uniformizable) if no topology without any isolated points stronger than τ is Tyconoff (or uniformizable).

Lemma 3.1. If U_{2} <^{⋆} U_{1} and V_{2} <^{⋆} V_{1}, then

(i) U_{2} ∧ V_{2} <^{⋆} U_{1} ∧ V_{1}; (ii) U_{2} ∧ V_{2} <^{⋆} U_{1} and U_{2} ∧ V_{2} <^{⋆} V_{1}.

Proof. (i) Let U_{2} ∩ V_{2} ∈ U_{2} ∧ V_{2}, where U_{2} ∈ U_{2}, V_{2} ∈ V_{2}. Then
St(U_{2} ∩ V_{2}; U_{2} ∧ V_{2})⊂ ⋃ {(U_{α} ∩ V_{β}) : U_{α} ∈ U_{2} with U_{α} ∩ U_{2} ≠ ϕ and
V_{β} ∈ V_{2} with V_{α} ∩ V_{2} ≠ ϕ}

⊂ ⋃ {U_{α} ∈ U_{2} : U_{α} ∩ U_{2} ≠ ϕ} ∩ ( ⋃ {V_{β} ∈ V_{2} : V_{β} ∩ V_{2} ≠ ϕ})

= St(U_{2}; U_{2}) ∩ St(V_{2}; V) ⊂ U_{1} ∩ V_{1} ∈ U_{1} ∧ V_{1}

[as U_{2} <^{⋆} U_{1} and V_{2} <^{⋆} V_{1}, so for U_{2} ∈ U_{2}, V_{2} ∈ V_{2} there exist some U_{1} ∈ U_{1} and some V_{1} ∈ V_{1} such that St(U_{2}; U_{2}) ⊂ U_{1} and St(V_{2}; V_{2}) ⊂ V_{1}]. So for U_{2} ∩ V_{2} ∈ U_{2} ∧ V_{2} there exists U_{1} ∩ V_{1} ∈ U_{1} ∧ V_{1} such that St(U_{2} ∩ V_{2}; U_{2} ∧ V_{2}) ⊂ U_{1} ∩ V_{1}. Hence

U_{2} ∧ V_{2} <^{⋆} U_{1} ∧ V_{1}.

(ii) We know that U_{1} ∧ V_{1} < U_{1} and U_{1} ∧ V_{1} < V_{1} and also we know that for three covers W_{1}, W_{2}, W_{3} of X, if W_{3} <^{⋆} W_{2} < W_{1}, then W_{3} <^{⋆} W_{1}. Hence the result follows. □

Lemma 3.2. For a Tychonoff space (X, τ), if μ is the collection of all normally open covers of (X, τ) and U is a normal isolator cover of (X, τ) with the corresponding normal sequence of isolator covers …U_{3} <^{⋆} U_{2} <^{⋆} U_{1} = U, then μ′_{1} = μ ∪ {U_{1}, U_{2}, …} ∪ {V ∧ U_{k} : k = 1, 2, …; V ∈ μ} forms a base for some uniformity on X which generates a stronger Tychonoff (uniformizable) topology τ_{μ′_{1}} on X such that (X, τ_{μ′_{1}} ) is T_{2} and contains no isolated points.

Proof. Here μ′_{1} = μ ∪ {U_{1}, U_{2}, …} ∪ {V ∧ U_{k} : k = 1, 2, …; V ∈ μ}. First we shall prove that for W_{1}, W_{2} ∈ μ′_{1}, there exists a W_{3} ∈ μ′_{1} such that W_{3} <^{⋆} W_{1} and

W_{3} <^{⋆} W_{2}.

Now μ is itself a base for the fine uniformity on X generating the topology τ on X. So for W_{1}, W_{2} ∈ μ, there obviously exists a W_{3} ∈ μ such that W_{3} <^{⋆} W_{1} and

W_{3} <^{⋆} W_{2}.

Also by Lemma 2.2, for U_{k}, U_{m} ∈ {U_{1}, U_{2}, …}, there exists an U_{l} ∈ {U_{1}, U_{2}, …}

such that U_{l} <^{⋆} U_{k} and U_{l} <^{⋆} U_{m}.

Now we will have to check it for four possible cases:

(i) W_{1} ∈μ and U_{k} ∈{U_{1}, U_{2}, …}; (ii) W_{1} ∈μ and W ∈{V ∧ U_{k} : k = 1, 2, …; V ∈ μ} ; (iii) W_{1}, W_{2} ∈ {V ∧ U_{k} : k = 1, 2, …; V ∈ μ}; (iv) for U_{k} ∈ {U_{1}, U_{2}, …} and W ∈ {V ∧ U_{k} : k = 1, 2, …; V ∈ μ}.

Case (i): Suppose W_{1} ∈ μ and U_{k} ∈ {U_{1}, U_{2}, …}. As μ is a base for the fine uniformity on X, there exists a W_{2} ∈ μ such that W_{2} <^{⋆} W_{1} and also U_{t} <^{⋆} U_{k} [where t is a positive integer such that t > k]. Hence by Lemma 3.1, W_{2} ∧ U_{t} <^{⋆} W_{1} ∧ U_{k}

and W_{2} ∧ U_{t} <^{⋆} W_{1}; W_{2} ∧ U_{t} <^{⋆} U_{k}, where W_{2} ∧ U_{t} ∈ μ′_{1}.

Case (ii): Let U′ ∈ μ, U ∧ U_{k} ∈ {V ∧ U_{l} : l = 1, 2, …; V ∈ μ}. Since U′, U ∈ μ and μ is being a base for the fine uniformity on X, there exists

a U′′∈ μ such that U′′<^{⋆} U′ and U′′<^{⋆} U … (I).

Now as U′′ <^{⋆} U and U_{t} <^{⋆} U_{k} (where t is a positive integer greater than k), by

Lemma 3.1, U′′∧ U_{t} <^{⋆} U ∧ U_{k}.

Again for U ∩ V ∈ U′′∧ U_{t}, we have St(U ∩ V; U′′∧ U_{t}) ⊂ St(U; U′′) ⊂ U′ (for

some U′ ∈ U′ as U′′<^{⋆} U′). So U′′∧ U_{t} <^{⋆} U′.

Case (iii): Let U′ ∧ U_{k}, U′′∧ U_{l} ∈ {V ∧ U_{t} : t = 1, 2, …; V ∈ μ}. Now, for

U′, U′′∈ μ, there exists a U′′′∈ μ, such that U′′′<^{⋆} U′, U′′′<^{⋆} U′′… (a).

Also Lemma 2.2 ensures that, for a positive integer t greater than both k and

l, U_{t} <^{⋆} U_{k} and U_{t} <^{⋆} U_{l} … (b).

From (a), (b) and Lemma 3.1, we get U′′′∧ U_{t} <^{⋆} U′∧ U_{k} as well as U′′′∧ U_{t} <^{⋆} U′′∧ U_{l}. Here we note that U′′′∧ U_{t} ∈ {V ∧ U_{t} : t = 1, 2, …; V ∈ μ} ⊂ μ′_{1}.

The proof of case (iv) can be done similarly. So we have for any W_{1}, W_{2} ∈ μ′_{1}, there exists a W_{3} ∈ μ′_{1} such that W_{3} <^{⋆} W_{1},

W_{3} <^{⋆} W_{2}. Hence μ′_{1} is a base for some covering uniformity on X. Now μ′_{1} generates the topology τ_{μ′_{1}} on X. Since μ ⊂ μ′_{1}, the topology generated by μ i.e., the topology τ is weaker than τ_{μ′_{1}} . Now {St(x; U) : U ∈ μ′_{1}} forms a local base at x ∈ X in (X, τ_{μ′_{1}} ) and also St(x; U) is infinite for each x ∈ X and for each U ∈ μ′_{1}. So (X, τ_{μ′_{1}} ) contains no isolated points. Also τ_{μ′_{1}} is T_{2} and uniformizable. Hence the Lemma follows. □

Theorem 3.1. For a Tychonoff space (X, τ), the following statements are equivalent:

(i) (X, τ) is maximal Tychonoff. (ii) Every normal isolator cover U of (X, τ) has an open refinement V, which

is also a cover of (X, τ). (iii) Every normal isolator cover U of (X, τ) has an open star refinement V,

which is also a cover of (X, τ).

Proof. We shall proceed to prove in the following manner: (i)⇔(ii), (ii)⇔(iii). (i)⇒(ii): Let (X, τ) be a maximal Tychonoff space and also let U be a normal

isolator cover of (X, τ) and U = U_{1}, U_{2}, … be the corresponding normal sequence of isolator covers of (X, τ).

Now we shall consider the collection μ′_{1} of covers consisting of all normally open covers of (X, τ), the covers {U_{1}, U_{2}, …} and the covers {V∧U_{k} : k = 1, 2, …; V ∈ μ}, where μ is the collection of all normally open covers of (X, τ).

So by Lemma 3.2, μ′_{1} is a base for some uniformity on X, which generates a stronger Tychonoff(or uniformizable) topology τ_{μ′_{1}} on X (i.e. τ_{μ′_{1}} ⊃ τ) such that (X, τ_{μ′_{1}} ) is T_{2} and contains no isolated points. Then τ = τ_{μ′_{1}} , by the maximality of τ as a Tychonoff (or uniformizable) topology. So we can write τ_{μ′_{1}} ⊂ τ.

Now {St(x; W) : W ∈ μ′_{1}} is a local base at x ∈ X in (X, τ_{μ′_{1}} ) and {St(x; W′) : W′∈ μ} is a local base at x ∈ X in (X, τ). By the Hausdorff criterion, for U_{2} ∈ μ′_{1}, there exists a V ∈ μ such that St(x; V) ⊂ St(x; U_{2}). Take a V_{x} ∈ V containing x. Then V_{x} ∈ St(x; V) ⊂ St(x; U_{2}) ⊂ St(U_{2}; U_{2}) [for some U_{2} ∈ U_{2} containing x]. As U_{2} <^{⋆} U_{1} = U, there exists some U_{x} ∈ U_{1} = U such that St(U_{2}; U_{2}) ⊂ U_{x}. So V_{x} ⊂ U_{x}. Therefore the cover W = {V_{x} : x ∈ X} is the required open cover of (X, τ), which is a refinement of U.

(ii) ⇒ (i): Let the condition holds and if possible let τ_{1} be a Tychonoff (or uniformizable) topology that contains no isolated points satisfying τ ⊂ τ_{1}. It is sufficient to prove that τ_{1} ⊂ τ.

Let U ∈ τ_{1}. Consider the collection μ_{1} of all normally open covers of (X, τ_{1}). Then for x ∈ U ∈ τ_{1}, there exists a U′∈ μ_{1} such that x ∈ St(x; U′) ⊂ U. But as U′ is a normal isolator cover of (X, τ_{1}), it is therefore so in (X, τ), as well. Hence the assumption shows the existence of an open cover W of (X, τ) satisfying W < U′. Obviously x ∈ St(x; W) ⊂ St(x; U′) ⊂ U and hence U ∈ τ as U is a neighborhood of x in (X, τ). So τ_{1} ⊂ τ and hence (i) is followed.

(ii)⇒(iii): Let (ii) holds i.e. every normal isolator cover of (X, τ) has a τ-open refinement. Let U be a normal isolator cover of (X, τ) with …, U_{3}, U_{2}, U_{1} = U be the corresponding normal sequence of isolator covers. Since U_{2} is also a normal isolator cover, by (ii), it has a τ-open refinement V. Now for V ∈ V, there exists a U_{2} ∈ U_{2}, such that V ⊂ U_{2}. So St(V; V) ⊂ St(U_{2}; U_{2}), as V < U_{2} and every member of V which intersects V must be contained in a member of U_{2} intersecting U_{2}. Since U_{2} <^{⋆} U, St(V; V) ⊂ St(U_{2}; U_{2}) ⊂ U for some U ∈ U. That is for V ∈ V, there exists a U ∈ U such that St(V; V) ⊂ U and hence V is a star refinement of U.

(iii)⇒(ii): As every star refinement of a cover W is obviously a refinement of W, (iii)⇒(ii) follows obviously. □

Corollary 3.1. In a maximal Tychonoff space (X, τ), a subset G of X is open if and only if for every x ∈ G there exists a normal isolator cover U of (X, τ) such that x ∈ St(x; U) ⊂ G

Proof. Let G be an open subset of (X, τ) and x ∈ G. Now we know that the collection of all normally open covers μ (say) of (X, τ) forms a base for a uniformity on X, which generates τ on X where β_{x} = {St(x; V) : V ∈ μ} forms a local base at x ∈ X in (X, τ). Hence the necessary part is followed as every normally open cover is also a normal isolator cover.

Conversely, let G be a subset of X such that for every x ∈ G there exists a normal isolator cover U of (X, τ) with x ∈ St(x; U) ⊂ G. Since (X, τ) is maximal Tychonoff, then by Theorem 3.1 the normal isolator cover U has an τ-open refinement V (say). Now if U is a member of V containing x, then obviously x ∈ U ⊂ St(x; U) and so x ∈ U ⊂ G. Hence G is a τ-neighborhood of x and this is true for every x ∈ G. Thus G is an open set in (X, τ). □

Basu and Mandal [2] recently characterized various disconnectedness in terms of star refinement of covers. Here using the tool of normal isolator covers, we have shown that a maximal Tychonoff (or maximal uniformizable) space is extremally disconnected.

Theorem 3.2. Let (X, τ) be a maximal Tychonoff space. Then for every nonempty proper subset G of X containing no isolated points in G (with the induced subspace topology), cl(G) is open.

Proof. Let G be a nonempty proper subset of X containing no isolated points in (X, τ). If cl(G) = X, then cl(G) is obviously open; so let cl(G) ⊊ X. Then X − cl(G) is an open set and obviously contains no isolated points in (X, τ). Also cl(G) obviously contains no isolated points in (X, τ).

Now we consider the cover U = {cl(G), X − cl(G)} of X. Here we see that for any normally open cover V of (X, τ) if x belongs to X − cl(G), then St(x; U ∧ V) being a union of open sets is itself an open set and hence is infinite. Let x ∈ cl(G). Now if x ∈ G, then x is not an isolated point of G. So when an open set containing x intersects G, it intersects at infinite number of points. Again if x is a limit point of G, then as (X, τ) is T_{1}, every open set containing x intersects G at infinite number of points. So in both cases St(x; U ∧ V) is infinite. Hence U is an isolator cover of (X, τ). Also …, U, U, U is a normal sequence and so U is a normal isolator cover of (X, τ). Thus by Theorem 3.1, U has a τ-open refinement. So cl(G) is obviously τ-open. Hence the result follows. □

L. Feng and S. Garcia-Ferreira, in their paper [12], have established that every maximal Tychonoff space is extremally disconnected [12, Lemma 1.6]. Here an alternative proof of this result is established.

Corollary 3.2. A maximal Tychonoff space is extremally disconnected.

Proof. As in a maximal Tychonoff space, every nonempty proper open set contains no isolated points in that space, so by Theorem 3.2, the closure of every open set is open and hence a maximal Tychonoff space is extremally disconnected.

□

Corollary 3.3. A maximal Tychonoff space is zero dimensional space.

Proof. The proof follows immediately, since an extremally disconnected regular space is zero-dimensional. □

Theorem 3.3. In a maximal Tychonoff space (X, τ), for any proper nonempty subset G containing no isolated points in (X, τ), cl(G) is maximal Tychonoff.

Proof. Let (X, τ) be maximal Tychonoff and G be a proper nonempty subset of X such that G contains no isolated points and also let cl(G) ≠ X. Then obviously (cl(G), τ_{cl(G)} ) is T_{2}, uniformizable and also contains no isolated points.

Now if U is any normal isolator cover of (cl(G), τ_{cl(G)} ), then U ∪ {X − cl(G)} is also a normal isolator cover of (X, τ).

As (X, τ) is maximal Tychonoff, by Theorem 3.1, U ∪{X −cl(G)} has a τ-open refinement V′, which is also a cover of X. Now V = V′_{cl(G)} = {V ∩ cl(G) : V ∈ V′} is an open cover of cl(G) and it refines U. So V is a τ_{cl(G)} -open refinement of U i.e., every normal isolator cover of (cl(G), τ_{cl(G)} ) has a τ_{cl(G)} -open refinement. Hence by Theorem 3.1, (cl(G), τ_{cl(G)} ) is a maximal Tychonoff space. □

Corollary 3.4. In a maximal Tychonoff space (X, τ), for every proper nonempty open set A, cl(A) is maximal Tychonoff.

Proof. As every proper nonempty open set contains no isolated points, the proof follows from Theorem 3.3. □

Corollary 3.5. A topological space (X, τ) is maximal Tychonoff if and only if every nonempty open subset is maximal Tychonoff.

Proof. The proof follows from Corollary 3.4. □

Acknowledgement. The authors would like to thank the learned referee for her valuable comments and suggestions.

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Department of Mathematics (Received 02 05 2013) West Bengal State University (Revised 21 05 2015 and 25 09 2015) Berunanpukuria, Kolkata, West Bengal India

    ckbasu1962@yahoo.com

Department of Mathematics Krishnagar Women’s College Nadia, West Bengal India

    snehadrisekharmandal@gmail.com