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JoannieM |
4:29pm on Sunday, October 24th, 2010 |

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rudgej |
9:10pm on Thursday, October 21st, 2010 |

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5:18am on Friday, September 24th, 2010 |

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10:07pm on Thursday, September 9th, 2010 |

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9:23pm on Sunday, September 5th, 2010 |

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4:09am on Monday, August 2nd, 2010 |

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DJHasis |
3:04pm on Wednesday, July 7th, 2010 |

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10:40am on Thursday, April 29th, 2010 |

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### Documents

ROCKY MOUNTAIN JOURNAL OF MATHEMATICS Volume 5, Number 2, Spring 1975

THE LATTICE OF TOPOLOGIES: A SURVEY

ROLAND E. LARSON AND SUSAN J. ANDIMA

In 1936, Garrett Birkhoff noted that inherent in the study of topology is the notion of the comparison of two different topologies on the same basic set. In his paper, On the combination of topologies, Birkhoff first explicitly described this comparison by ordering the family of all topologies on a given set, and looking at the resulting lattice. BirkhofFs ordering was the natural one of set inclusion; that is, if U and ' are both topologies on a given set, is less than or equal to ' iff U is a subset of D ''. The topological definitions of this paper coincide mainly with those of W. J. Thron as found in Topological Structures, and the lattice definitions follow Garrett BirkhofFs Lattice Theory. However, for the reader's convenience we will begin with some lattice-theory preliminaries, as well as listing several of the less familiar topological definitions in the glossary. In order to make reading easier, a consistent notation will be used throughout the paper; however, it should be noted that in some of the entries in the bibliography, the lattice considered is the dual of the lattice of topologies to which we refer. This survey is divided into five sections. The first two sections deal with the lattice of topologies and the lattice of Tx-topologies. The third section contains short summaries of lattices of different subsets of the set of topologies, while the fourth section summarizes lattices of structures which contain, or partially contain, the set of topologies. The final section discusses minimal and maximal topologies. Before proceeding, we should remark that we have consciously omitted some of the more detailed results. Furthermore, in a survey of this nature, we have surely overlooked work which should have been included. We gladly welcome any additions or corrections. Lattice Terminology and Notation. A partially ordered set, (L,^) is a set L on which a binary relation ^ has been defined. The relation is reflexive, anti-symmetric and transitive. A partially ordered set (L, ) is called a lattice if any two elements in the set have a greatest lower bound or meet denoted by a A b and a least upper bound or join denoted by a V b. A lattice is called complete if any of its subsets have a meet and join in the set.

Received by the editors February 17, 1972 and in revised form October 2, 1972.

Copyright 1972 Rocky Mountain Mathematics Consortium

R. E. LARSON AND S. J. ANDIMA

(L, ^ ) is called the dual of the lattice (L, ^ ). (A, ^ ) is called a sublattice of (L, ^ ) if A L and finite meets and joins are preserved. (A, ^ ) is called a complete sublattice of (L, = ) if arbitrary meets and joins are preserved. By "a covers b" in a lattice (L, = ) we mean b^ a and b^ c^ a implies b = c or c = a. The teas element of a lattice is designated O and the greatest element is designated L An atom is an element which covers the least element. A lattice is atomic if every element other than O can be written as the join of atoms. An anti-atom is an element which is covered by L A lattice is antiatomic if every element other than / can be written as the meet of anti-atoms. An element a is called the complement of b in a lattice if a A b = O and a V b = I. A lattice is called complemented if every element has at least one complement, uniquely complemented if every element has exactly one complement. A lattice is called distributive if a A (b V c) = (a A b) V (a A c) and a V (b A c) = (a V b) A (a v c) for all a, b, c in the lattice. A lattice is called modular if a = c implies a v (b A C) = (a v b) Ac. A lattice L is called upper semi-modular iff for distinct a and b in L such that a and b both cover c, then a M b covers both a and b. Lower semi-modular is defined dually. If L is a complete atomic lattice with the set of atoms A, then L is called tall iff for every PC. A, where p = V {a | a G ? } , {a | a G A, a ^ p) = Pi {B | F B A, a, bGB and c^a v b implies cB}, A map from a lattice L to a lattice K is called a lattice homomorphism if it preserves finite meets and joins. The map is called a complete homomorphism if it preserves arbitrary meets and joins. A lattice isomorphism is a lattice homomorphism which is one to one and onto. A lattice (L, = ) is called self dual if it is lattice isomorphic to

THEOREM 1.5. If |X| = 1, 2, 3, 4, 5, 6, or 7, |J(X)| = 1, 4, '29, 355, 6,942, 209,527, or 9,535,241. If |X| = n / 1, 2" ^ \X(X)\ g 2^~l\ [29], [35], [62]

The cardinality of 2(X), when X = 5 or 6, is listed incorrectly in [97].

The Irregular Lattice Structure of 2(X). The following three theorems help to point out the complex structure of 2(X).

THEOREM 1.6. If\X\ > 2, 2(X) is non-distributive, non-modular, and neither upper nor lower semi-modular. [ 117], [ 100], [67]

Let X = {a,b,c} and let ^U be the principal ultrafilter of a. The following diagram from [67] illustrates Theorem 1.6.

{0,{a},{b},{a,b},X} {0,{a},{a,c},{a,b},X}

#### {0,{a},{a,b},X}

It is interesting to note that if X is finite and contains more than three elements, that there are more atoms in %(X) than anti-atoms. However, if X is infinite, the reverse is true. This points out the following theorem.

#### THEOREM THEOREM

1.7. If\X\ > 3, (X) is not self-dual 1.8. 2(X) is tall iffX is finite. [41]

Lattice Embeddings in 2(X).

THEOREM 1.9. For any lattice L, there exists a set X, such that L may be embedded in 2(X).

This is an extension of Whitman's well-known result that any lattice may be embedded in the lattice of partitions on some set. [123] Since the lattice of all partition topologies on X forms a complete sublattice of 2(X), [117] and since the lattice of partition topologies on X is isomorphic to the dual of the lattice of all partitions on X, [89] the result follows. Morphisms of 2(X).

THEOREM 1.10. If \X\ ^ 2, 2(X) has only trivial lattice homomorphisms. That is, any lattice homomorphism of X(X) onto a lattice L, is either a lattice isomorphism or L consists of a single element. [41] THEOREM 1.11. If X contains one or two elements, or X is infinite, the group of lattice automorphisms of 2(X) is isomorphic to the sym-

metric group on X. IfX is finite and contains more than two elements, the group of lattice automorphisms of 2(X) is isomorphic to the direct product of the symmetric group on X with the two element group. [41], [37] In the proof of Theorem 1.11, Hartmanis used the atomic structure of 2(X). Later, Frhlich proved the same theorem using the antiatomic structure of 2(X). Theorem 1.11 has the following consequence. If X is an infinite set and F is any topological property, then the set of topologies in 2(X) possessing property P may be identified simply from the lattice structure of 2(X). This follows from Theorem 1.11 since the only lattice automorphisms of 2(X) for infinite X are those which simply permute the elements of X. Therefore, any automorphism of 2(X) must map all the topologies in 2(X) onto homeomorphic images. Thus the topological properties of elements of 2(X) must be determined by the position of the topologies in 2(X). An example is shown in the following theorem. 1.12. If U is an anti-atom in 2(X), then is T1 iff O possesses no maximum complement in 2(X). [96]

2.7. A(X) is not modular, and hence not distributive.

The following example to illustrate Theorem 2.7 is taken from [67]. Let X be an infinite set such that A and X ~ A are both infinite. Let ^ be the cofinite topology on X and choosing x (f A, let gx= g\j {0,A,X} ^ 2 = ^ V {0,AU{x},X} ^ 3 = 9 V { 0 , {x},X} g4= g\J { 0 , {*},A,AU{s},X}.

The following diagram is valid.

2.8. A(X) is both upper and lower semi-modular. [67]

THEOREM 2.9. Any nontrivial interval in A(X) contains a covering relation. [68]

Lattice Embeddings in A(X). THEOREM 2.10. If Lis any finite distributive lattice, there exists a set X and topologies U and ' in A(X) such that L is isomorphic to the interval between O and O '. [68] Morphisms of A(X).

THEOREM 2.11. If X is infinite, the group of automorphisms A(X) is isomorphic to the symmetric group on X. [34]

THEOREM 2.12. If X is infinite, the lattice of complete homomorphisms of A(X) is isomorphic to the lattice of finite subsets of X and the set X ordered under set inclusion. [41]

III. Lattices of Subfamilies of 2(X). Before proceeding with this section, we present the following chart which shows how several

topological properties are preserved under lattice operations in 2(X). For proofs or references, see [39]. In the chart, we will use the following notation: ^ A A ^ V v + preservation under weakening of topologies. preservation under arbitrary meets. preservation under finite meets. preservation under strengthening of topologies. preservation under arbitrary joins. preservation under finite joins. indicates that the topological property is preserved.

Property Tx T 0 , TD, T2, and totally disconnected 1^3, T3a, regular, completely regular, and zero-dimensional 1st and 2nd countable principal bicompact, Lindelf, [connected, and separable locally connected T4, T5, normal, completely normal, paracompact, and |locally bicompact

Preservation under lattice operations ^ A A ^ V v

This listing points out that of the above common properties, the Tx separation axiom holds a special place, in that it is preserved under arbitrary meets and joins in 2(X). Of course, subfamilies of 2(X) may form lattices under set inclusion even if they do not preserve arbitrary meets and joins in 2(X). We present five such examples.

The Lattice of Principal Topologies. A topological space is called principal if it is discrete or if it can be written as the meet of principal ultratopologies. A. Steiner proved that this is equivalent to requiring that the arbitrary intersection of open sets is open. While A(X) possesses a lattice structure which is quite different from 2(X), the lattice of principal topologies compares closely as a lattice with 2(X). Most of the results concerning this lattice are due to A. Steiner. [100] The lattice of principal topologies is a complete lattice whose least element is the indiscrete topology and greatest element is the discrete topology. This lattice is both atomic and anti-atomic; its atoms coincide with those of 2(X). Although the lattice of principal topologies is a sublattice of (X), it is not a sub-complete lattice of 2(X). This lattice is complemented, non-modular if |X| = 3, and non-self dual if |X| = 4. If X is finite, the lattice of principal topologies is simply 2(X). If X is infinite, the lattice of principal topologies is of cardinality 2lxl. The lattice of principal topologies on X is isomorphic to the dual of the lattice of pre-orders on X. [ 100] Alexandroff pointed out this correspondence between orders and topologies when he proved that there is a one-to-one correspondence between the principal T0topologies on X and the partial orders on X. [ 1] For more information on this subject, see [ 124]. The Lattice of Partition Topologies. A topology on X is called a partition topology if it possesses a base which is a partition of X. The lattice of all partition topologies on a set has been studied in several different forms. O. Ore seems to have been the first to extensively investigate this lattice in the form of the lattice of equivalence relations on a set. [78] Vaidyanathaswamy pointed out that the lattice of partition topologies is a sublattice of 2(X). [117] More recently, M. Rayburn studied this lattice as the lattice of closed-open topologies and as the lattice of complete Boolean Algebras on X. [89] Finally, M. Huebener has proved that the lattice of partition topologies on X is precisely the lattice of principal-regular topologies on X. [46] The least and greatest elements of this lattice are the indiscrete and discrete topologies. Ore has shown that this lattice is completely complemented; that is, every interval in the lattice is a complemented lattice. He also proved that this lattice is atomic and anti-atomic, where the atoms are the partition topologies containing precisely two minimal open sets, and the anti-atoms are the partition topologies

which have one minimal open set consisting of a pair of points, and the other minimal open sets each consisting of a single point. The lattice of partition topologies on X is lower semi-modular, but non-modular if |X| ^ 4. If X is infinite, the cardinality of this lattice is 2lxL If |X| = n, the cardinality of the lattice of partition topologies on X, denoted p n , is given by the recursion formula

#### Pn+1= S ( ? V i

Ore also proved that the group of automorphisms of the lattice of partition topologies on X is isomorphic to the symmetric group on X. [78] As a final comment, we point out, once again, Whitman's result that any lattice may be embedded in the lattice of partition topologies on an appropriate set. [ 123] The Lattice of Regular Topologies and the Lattice of Completely Regular Topologies. The join of regular (completely regular) topologies in X(X) is itself a regular (completely regular) topology. However, this is not necessarily true of the meet of regular (completely regular) topologies. [43] [75] Hence, neither of these lattices is a sublattice of the lattice of topologies. The least and greatest elements of each lattice are the indiscrete and discrete topologies. If X is a finite set, the lattice of regular topologies on X is precisely the lattice of partition topologies on X. [46] M. Hu ebener has described certain classes of regular topologies which have complements in the lattice of regular topologies, [46] but it is apparently unknown if the entire lattice is complemented. Since the non-principal ultratopologies in 2(X) are both regular and completely regular, [100] the cardinality of both lattices on an infinite set is 2 2. The Lattice of Countably Accessible Topologies. This lattice was introduced by R. Larson [66] in a construction which paralleled A. Steiner's construction of the lattice of principal topologies. A topology is called a countably accessible topology if it can be written as the meet of ultratopologies whose associated ultrafilters contain countable sets, but no finite sets. A topology is countably accessible if and only if every non-closed set, G, contains a countable subset with a limit point lying outside of G. The set of countably accessible topologies on a set is a subset of the collection of Tx -topologies and contains all first countable T r topologies on the set. Arbitrary meets in this lattice and (X) coincide, although arbitrary joins may differ. The lattice is non-atomic, anti-

atomic, non-complemented, and non-modular. If X is countable, the lattice of countably accessible topologies is precisely A(X). If X is uncountable, the cardinality of the lattice of countably accessible topologies is 2'xl. The Lattice of "Joins of Hyperplanes". R. Bagley defined this lattice in [7] to be the lattice consisting of the cofinite topology, the atoms in A(X), and any topology which can be written as the join of atoms in A(X). This lattice is a sub-complete lattice on A(X), and is isomorphic to the complete Boolean algebra of subsets of X. IV. 2(X) is contained, in a natural way, in several more general lattices, which arise from considering some of the alternative approaches to topological structure through closure functions or convergence functions. Lattices of Closure Functions. A closure function in the sense of Cech is an increasing, order-preserving, function / from F(X) to F(X) such that f(0) = 0. Ore's definition adds to these the property that / is idempotent. If, in addition, the function preserves unions, as is assumed in the Kuratowski axioms, the structure is equivalent to the usual topological structure. When ordered by f^ g iff g(A) C f(A) for all A Q X, the set of Cech-closures and the set of Ore-closures on X both form complete lattices containing the lattice of topologies on X. [61], [77] The lattice of Cech-closures has the virtue of simple joins and meets defined by ( / v g)(A) = f(A) H g(A), and ( / A g)(A) = f(A) U g(A) for all A C X. It is a proper sublattice of the lattice of all functions from (P(X) to ^ ( X ). Unfortunately, neither the lattice of Ore-closures nor the lattice of topologies is a proper sublattice of the lattice of Cech-closures. The Ore-closures have the same join as the Cech-closures, but different meet. The lattice of topologies has the same meet as the lattice of Ore-closures, but a different join. [61], [77] The lattice of Ore-closures can be represented as the lattice of complete intersection rings over X. It is a lower semi-modular atomic lattice, but is not anti-atomic and not complemented. [77] Lattices of Sequential Topologies. When Garrett Birkhoff introduced the lattice of topologies, he also discussed the lattice of sequential topologies. In its most general form, a sequential topology on a set X is just a relation between the set of sequences on X and X itself. If / is a sequence topology on X, then a sequence {xn} converges to a point x if x E. f({xn}). We denote this as xn>x. A Frechet

./-space is a sequential topology that satisfies: (1) f({xn}) is empty or singleton. (2) If x = xn for every n, then xn x. (3) xn > x implies that every subsequence of {xn} converges to x. The sequential topologies, when ordered by f= g iff g({x n }) f({xn}) form a complete, distributive lattice, of which the lattice of Frech et./-spaces is a sub-complete lattice. [19] If the Frechet spaces are further assumed to satisfy the condition that the addition of a finite number of terms to a sequence affects neither its convergence nor its limits, then the corresponding lattice is complete and completely distributive. [117] These lattices do not, however, contain 2(X) in any natural way. The lattice of topologies and the lattice of sequential topologies can be mapped into each other by the functions f and i/i, defined by (^)({ x n}) = {x | every open neighborhood of x in U contains all but a finite number of the xn} and ijj(f) = {A\ AC X, f({xn}) C A implies that A contains all but a finite number of the x n }. It is clear that, for any topology U, I/J((^7)) is always TY. Furthermore, even when O is T1? */>((^)) does not always equal D. [117] Thus fails to include, not only the lattice of topologies, but also the lattice of 7\-topologies, in the lattice of sequential topologies. A satisfactory topological characterization of the topologies for which </*(( ^ ) ) = *J, and thus of the class of topologies which can be included in the lattice of sequential topologies on X, has apparently not been determined. Lattices of Convergence Structures. Convergence structures generalizing topological structure can be found by using convergences, not of sequences, but of filters, or alternatively, of nets. The definitions and notation used here are those of Kent. [50], [51] Let F(X) be the set of all filters on X, and let Qx be the filter generated by x, for each x G X. A convergence function / is a function from F(X) into iP(X), suchthat (1) V g ^>f(V)f(g), for all <3, ^ E F(X), and (2) x G / ( ^ x ) , f o r a l l x G X. For each x G X, let O/fa) = Pi {<? | <? is a filter and / ( 9 ) }. A series of progressively stronger structures, culminating in one equivalent to topological structure, follows. A convergence function f is a convergence structure iff

#### (3)xGf(V)^xEf(Vn

A convergence structure fis a limitierung [36] iff (4) / ( <?0 fi / ( <?2) / ( S , fi 92), for all Vlt G F(X). A limitierung / is a pseudo-topology iff ( 5 ) i G / ( 9 ' ) for all ultrafilters S S? '=>x G / ( S?). A pseudotopology / is a pretopology iff (6) x G f(0//x))9 for all x E X.

A pretopology is topological iff (7) for every x G X, ^y{x) has a filter base iBy(x) .<V/x) such thatt/ G G(x) G Q/j(x)=>G(x) % ).

For each of these properties, the set of all such functions on a fixed set X forms a complete lattice when ordered in the natural way by / i ^ h iSM * ) e /x( *), for all <? G F(X). The lattice of pretopologies on X has a representation as a sublattice of filters on X x , and, as a consequence, is atomic, anti-atomic, modular, distributive, and compactly generated, but not completely distributive, and not complemented unless X is finite, in which case it is uniquely complemented. [26] Carstens has stated that the lattice of pseudotopologies on X can be represented as the lattice of subsets of a set and is therefore a complete Boolean lattice. [27] The lattice of convergence structures, C(X), is a sub-complete lattice of the lattice of convergence functions, C'(X). Both C(X) and C'(X) have join and meet that can be defined very simply for any family Q of functions as ( V ) ( 9 ) = f l {/(S?) | / } , and (A Q)(V) = U { / ( S ) | / G ) } , for all VGF(X). Each of the other lattices is an additive subsystem of C'(X), in that it has the same join, both finite and infinite. But none is a sub-complete lattice of C'(X) because meets are not preserved. In fact, Kent has shown that every convergence structure is the infimum of a set of topologies. [51] Therefore, for any two lattices intermediate between 2(X) and C'(X), one can never be a sub-complete lattice of the other. However, Carstens has shown that (X) is a sublattice of the lattice of pretopologies on X. For each convergence function f, there is a finest limitierung, a finest pseudo-topology, a finest pretopology, and a finest topology coarser than, or equal to /. [36], [50] V. Strengthening of Topologies. If *J and ' are topologies on X such that O C C ', O ' has been called an expansion, [43] an exten7 sion, [73], [21] and an enlargement of *J. [30] Most of the well

References [81], [64], [39] [124] [64], [39] [19], [13]

#### Tg and T{, TD T,

T2 \T2

#### [48], [86] [22], [13]

[45], [91]

^3a> T4, T 5 , minimal iff it possesses the property T 2 -paraunder consideration and is bicompact. compact, Metrizable, T2-locally compact, T2-zero |dimensional

#### [13], [45] [98], [105]

T2-perfectly minimal iff it is T2-perfectly normal |normal and countably compact. T2-first countable,

minimal iff T2 -first countable and every [105], [83] open filter with a countable base and a unique cluster point converges. minimum iff X is countable and *J is the minimum Tx topology. [83]

#### GLOSSARY

Topological Definitions: A space (X, U ) is called a TD-space iff {x} ' (the derived set of {x}) is a closed set for every x G X. Tg-space iff whenever {x} ' ^ 0 , {x} ' is a point closure. Ti -space iff for each x G X, {*}' is the union of a family of pointclosures, {{y} | y G Y} such that for all distinct r, s G Y, r and s are separated. T 2o -space iff it is a Urysohn space. T 3 -space iff it is Tx and regular. T3fl-space iff it is T\ and completely regular. T 4 -space iff it is Tx and normal. T 5 -space iff it is Tx and completely normal. E 0 -space iff every point in X can be written as the countable intersection of neighborhoods of x. EY -space iff every point in X can be written as the countable intersection of closed neighborhoods of x. A filter in a space (X, C7) is called an open filter iff it has a filter base consisting of open sets. A filter in a space (X, ) is called a closed filter iff it has a filter base consisting of closed sets. A filter in a space (X, (J) is called a regular filter iff it is both an open and closed filter. A filter, 9 in a space (X, U) is called a Urysohn filter iff it is an open filter and for each x G X, such that x is not a cluster point of *?, there is an open neighborhood U of x and V 9 such that U DV

#### BIBLIOGRAPHY

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3. Bruce A. Anderson, Families of mutually complementary topologies, Proc. A.M.S. 29 (1971), 362-368. 4. , A class of topologies with T\-complements, (to appear in Funda. Math.) 5. C. E. Aull, A certain class of topological spaces, Prace Math. 11 (1967), 49-53. 6. R. W. Bagley, The topolattice and permutation group of an infinite set, Ph.D. dissertation, Univ. of Florida, 1954. 7. , On the characterization of the lattice of topologies, J. London Math. Soc. 30 (1955), 247-249. 8. R. W. Bagley and David Ellis, On the topolattice and permutation group of an infinite set, Math. Japon. 3 (1954), 63-70. 9. V. K. Balachandran, Minimal Bicompact spaces, J. Ind. Math. Soc. 12 (1948), 47-48. 10. , On the lattice of convergence topologies, J. Madras Univ. B28 (1958), 129-146. 11. B. Banaschewski, ber zwei Extramaleigenschaften topologischer Rume, Math. Nachr. 13 (1955), 141-150. 12. Manuel P. Berri, Minimal topological spaces, Ph.D. dissertation, Univ. of California at Los Angeles, 1961. 13. , Minimal topological spaces, Trans. Amer. Math. Soc. 108 (1963), 97-105. 14. , The complement of a topology for some topological groups, Fund Math. 58 (1966), 159-162. 15. , Categories of certain minimal topological spaces, J. Austral. Math. Soc. 3 (1964), 78-82. 16. Manuel P. Berri, Jack R. Porter and R. M. Stephenson, Jr., A survey of minimal topological spaces, presented at the Proceedings of the Indian Topological Conference in Kanpur, October, 1968, General Topology and its Relations to Modern Analysis and Algebra III, Academic Press (1970), 93-114. 17. Manuel P. Berri and R. H. Sorgenfrey, Minimal regular spaces, Proc. Amer. Math. Soc. 14 (1963), 454-458. 18. Garrett Birkhoff, Lattice theory, Amer. Math. Soc. Colloquium Pubi, third edition, Rhode Island, 1967. 19. , On the combination of topologies, Fund. Math. 26 (1936), 156166. 20. , Sur les espaces discrets, C. R. Acad. Sci. Paris 201 (1935), 19-20. 2 1. Carlos J. R. Borges, On extensions of topologies, Canad. J. Math. 19 (1967), 474-487. 22. Nicholas Bourbaki, Espaces minimaux et espaces compltement spars, C. R. Acad. Sci. Paris 212 (1941), 215-218. 23. Douglas E. Cameron, Maximal and minimal topologies, Ph. D. dissertation, Virginia Polytechnic Institute (1970). 24. , Maximal and minimal topologies, Trans. Amer. Math. Soc. 160 (1971), 229-248. 25. , Maximal pseudocompactness, Proceedings of Conference on General Topology, Emory Univ., 1970, 26-31. 26. Allan M. Carstens, The lattice of pretopologies on an arbitrary set S, Pacific J. Math. 29 (1969), 67-71. 27. , The lattice of pseudotopologies on S, Amer. Math. Soc. Notices 16 (1969), 150.

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R. E. LARSON AND S. J. A N D I M A

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PENNSYLVANIA STATE UNIVERSITY, BEHREND COLLEGE, E R I E , PENNSYLVANIA C. W. POST COLLEGE, LONG ISLAND UNIVERSITY, GREENVALE, N E W YORK

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