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The rudiments of the theoly of Galois connections (or residuation theory, as it is sometimes called) are provided, together with many examples and applications. Galois connections occur in profusion and are well known to most mathematicians who deal with order theory; they seem to be less known to topologists. However, because of their ubiquity and simplicity, they (like equivalence relations) can be used as an effective research tool throughout mathematics and related areas. If one recognizes that a Galois connection is involved in a phenomenon that may be relatively complex, then many aspects of that phenomenon immediately become clear, and thus, the whole situation typically becomes much easier to understand. Mathematics SubjectClassification. Primary 06A15, 06-01, 06A06; Secondary 54-01, 54B99, 54899,68F05. Key words and phrases. Galois connection, closure operation, interior operation, polarity, axiality. 'Galois' main results were published fourteen years after his early death (at the age of 21 in a duel) by Liouville in his Journal de mathimatiquespures et applqukes (1846). For a translation of Galois' original notes Memoire sur les conditions de rksolubiliti des iquationspar radicaux, see the text by Edwards [ 141. 103 104 ANNALS NEW YORK ACADEMY OF SCIENCES the world of intermediate fields of a field extension E : F and the world of subgroups of the group of automorphisms of E that fix the subfield F (cf. Examples 1 and 19).Today this area is known as Galois theory.Since the proofs of many results in this paper are either well known or easily obtained, we do not include them. Galois connections were originally expressed in a symmetric but contravariant form with transformations that reverse (rather than preserve) order. Early references to this form are [8], [22], [44], and [45]. We use the covariant form since it is more convenient, for example, compositions of Galois connections are handled more easily; it allows for more natural categorical explanations (e.g., by means of adjunctions); and it is more applicable to computer science situations (where relative information preservation is important). For references to the covariant form, see [7], [51], [9], [32], [25], [35], [15], [42], [31], and [19]. THE DEFINITION AND SOME OF ITS CONSEQUENCESWe formulate all our results in terms of partially ordered sets (orposets), that is, sets equipped with a reflexive, transitive, and antisymmetric relation. Everything can easily be generalized to preordered sets (i.e., one may drop the antisymmetry requirement) and even to preordered classes. Applications of these generalizations can be found in Examples 10, 22, and 23, and in [lo]. For a poset 9 = (P, s ) , the order-theoretical dual (P, 2 ) is denoted by 9 OP. =* DEFINITION 1: Consider posets 9 = (P, I) and d = (Q, E). If P -+ Q and Q 2 P are functions such that for allp E P and all q E Q p s a * ( q ) iff a * @ )~q (1)then the quadruple IT = (9, IT *, IT*, d ) is called a Galois connection. We also write 9 5 d (or sometimes just ( I T * , T*)) for the whole Gal...

a r t i c l e i n f o a b s t r a c tArticle history: Dedicated to Bob Lowen, who opened an easy approach to wonderful spaces MSC: primary 54D05 secondary 06B35, 06D10, 54D45, 54F05 Keywords: (Local) base (Co)frame Distributivity Hypercompact Locally connected Specialization order Strongly connected Supercompact Web space Various local connectedness and compactness properties of topological spaces are characterized by higher degrees of distributivity for their lattices of open (or closed) sets, and conversely. For example, those topological spaces for which not only the lattice of open sets but also that of closed sets is a frame, are described by the existence of web neighborhood bases, where webs are certain specific path-connected sets. Such spaces are called web spaces. The even better linked wide web spaces are characterized by F -distributivity of their topologies, and the worldwide web spaces (or C -spaces) by complete distributivity of their topologies. Similarly, strongly locally connected spaces and locally hypercompact spaces are characterized by suitable infinite distributive laws. The web space concepts are also viewed as natural extensions of spaces that are semilattices with respect to the specialization order and have continuous (unary, binary or infinitary) semilattice operations.

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We investigate the numbers $d_k$ of all (isomorphism classes of) distributive lattices with $k$ elements, or, equivalently, of (unlabeled) posets with $k$ antichains. Closely related and useful for combinatorial identities and inequalities are the numbers $v_k$ of vertically indecomposable distributive lattices of size $k$. We present the explicit values of the numbers $d_k$ and $v_k$ for $k < 50$ and prove the following exponential bounds: $$ 1.67^k < v_k < 2.33^k\;\;\; {\rm and}\;\;\; 1.84^k < d_k < 2.39^k\;(k\ge k_0).$$ Important tools are (i) an algorithm coding all unlabeled distributive lattices of height $n$ and size $k$ by certain integer sequences $0=z_1\le\cdots\le z_n\le k-2$, and (ii) a "canonical 2-decomposition" of ordinally indecomposable posets into "2-indecomposable" canonical summands.

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