Online Appendix: Generalized Topologies

Transcription

1 Supplementary Electronic Materials for Genotype-Phenotype Maps Peter F Stadler and Bärbel MR Stadler Biological Theory (3) 2006, Online Appendix: Generalized Topologies In the last years there has been increased interest in applications of generalized topological spaces also in other areas, including pattern recognition, image analysis, information representation, and artificial chemistry, see e.g. Eckhardt and Latecki (1994); Smyth (1995); Pfaltz (1996); Galton (2000); Marchand-Maillet and Sharaiha (1997); Largeron and Bonnevay (2002); LeBourgeois et al. (1999); Speroni di Fenizio et al. (2002); Dittrich and Speroni di Fenizio (2006) for a few examples. Topology textbooks, e.g. Gaal (1964); Steen and Seebach (1970), usually start out by defining a topology on a set X by means of a collection O P(X) of open sets. (The symbol P(X) denotes the power set, i.e., the set of all subsets of X.) Equivalently, the collection C of closed sets can be used. By definition, A C if and only if X \A O, i.e., A is an open set if its complement is a closed set and vice versa. In a topological space (X, C) the following three axioms hold: (I1) X C, i.e., the basis set is closed. (I2) For every index set I holds: If A i C for all i I, then i I A i C, i.e., arbitrary intersections of closed sets are closed again. (I3) If A,B C then A B C, i.e., finite unions of closed sets are closed. The corresponding axioms for open sets are obtained by exchanging unions and intersections. Setting I = /0 in (I2), we see that /0 C. We shall see below, however, that accessibility implies a structure that is weaker than a topology, i.e., we will not be able to argue that all tree of the above axioms should hold. In fact, open or closed sets do not seem to arise naturally in the context of either genetic or phenotypic accessibility. We therefore need to investigate mathematical structures that satisfy only a subset of the above axioms. In lattice theory so-called intersection structures are considered. These objects fulfill only the axiom (I2), see e.g., Davey and Priestley (1990). Intersection structures that in addition satisfy (I1) are called topped. Given an intersection structure (X,C), a closure function cl : P(X) P(X) can be defined that associates with each

2 set A X the smallest closed set containing A: cl(a) = {B C A B}. (1) The function cl has two important properties: (i) it is isotone, i.e., A B implies cl(a) cl(b) and (ii) it is idempotent, i.e., cl(cl(a)) = cl(a). If (I1) also holds, the closure function is expanding, i.e., A cl(a). In a topology, i.e., a topped intersection structure satisfying (I3), finally, the closure function is additive, i.e., cl(a B) = cl(a) cl(b). The notion of a closure function cl is closely related with two other basic concepts of point set topology: The conjugate or dual of the closure function is the interior defined by int(a) = X \ cl(x \ A), i.e., cl(a) = X \ int(x \ A). A set N is called a neighborhood of a point x X if x lies in the interior of N. This defines the neighborhood function N : X P(P(X)) as N (x) = { N X } x int(n) (2) The concept of neighborhoods makes sense also for sets: N is a neighborhood of A if and only if N is a neighborhood of each point of A, i.e., N (A) = x AN (x) (3) The notion of a boundary can also be derived from the closure function: A = cl(a) cl(x \ A) = cl(a) \int(a) (4) A topological theory based on boundaries was developed e.g. in Albuquerque (1941). From the boundary A we can recover closure and interior functions as cl(a) = A A and int(a) = A \ A. The biological interpretation of the boundary approach is discussed in some detail in Stadler et al. (2001). Closure, interior, and neighborhood are equivalent constructions on a set X, independent of any particular properties of cl, int, and N. Indeed, given the neighborhoods of each point x X it is possible to obtain the associated closure and interior functions (Day 1944): x cl(a) iff (X \A) / N (x) and x int(a) iff A N (x). Properties of the closure function cl can be translated into properties of the interior and neighborhood functions, and vice versa. Table 1 summarizes the basic axioms in all three languages. Instead of starting with systems of open or closed sets one can therefore build a topological theory directly on the closure functions; for instance Čech s book (1966) shows that many of the classical results of topology hold already for pretopologies that satisfy only (K0), (K1), (K2), and (K3) but lack idempotency (K4) of the closure function. This approach was pioneered by Day (1944), Hammer (1962; 1967) and Gniłka (1994). Surprisingly, meaningful topological concepts can already be defined on a set X endowed with an arbitrary set-valued set-function cl. Almost all approaches to extending the framework of topology at least assume the isotony axiom (K1), see e.g. Brissaud (1975); Day Biological Theory 1(3) 2006, Supplementary Electronic Materials 2

3 Table 1. The basic axioms for extended topological spaces. closure interior neighborhood (K0) cl(/0) = /0 int(x) = X N (x) (K1) A B = cl(a) cl(b) A B = int(a) int(b) N N (x), N N = N N (x) cl(a) cl(b) cl(a B) int(a) int(b) int(a B) cl(a B) cl(a) cl(b) int(a B) int(a) int(b) (K2) A cl(a) int(a) A N N (x) = x N (K3) cl(a B) cl(a) cl(b) int(a) int(b) int(a B) N,N N (x) = N N N (x) (K4) cl(cl(a)) = cl(a) int(int(a)) = int(a) N N (x) int(n) N (x) (K5) cl( i I A i ) = i I cl(a i ) cl( i I A i ) = i I int(a i ) {N N (x)} N (x) (1944); Gniłka (1994); Hammer (1955, 1962); Hausdorff (1935) and many others. The importance of isotony is emphasized by numerous equivalent conditions, some of which are listed in Table 1. If (K1) holds, we recover also a more familiar relationship between closure and neighborhood: cl(a) = { x X N N (x) : N A /0 } (5) In this case we may also express (K4) in a much more familiar form: Every neighborhood N of a point x contains an open neighborhood. Separation properties play an important role in classical point set topology. Examples are the following axioms (T0) For all x y there is a neighborhood N N (x) such that y / N or a neighborhood N N (y) such that x / N. (T1) For all x y exists N N (x) such that y / N. (T2) For all x y exists N N (x) and N N (y) such that N N = /0. (T2 1 2 ) For all x y exists N N (x) and N N (y) such that cl(n ) cl(n ) = /0. In neighborhood spaces, which satisfy (K0), (K1), and (K2), we still have the familiar implications between these properties: (T2 2 1 ) = (T2) = (T1) = (T0). So-called higher separation properties in neighborhood spaces are considered e.g. in Thampuran (1971); Stadler and Stadler (2003). We shall see below that even weaker notions of separation than the (T0)-axiom, known as thinness in the context of binary relations (McKenzie 1971), play an important role: (TH0) For all x y there is a z X and a N N (z) such that for all N N (z) satisfying N N holds x N and y / N or y N and x / N. (TH1) N (x) = N (y) implies x = y. Biological Theory 1(3) 2006, Supplementary Electronic Materials 3

4 It is not hard to check that in neighborhood spaces the (T0) separation axiom implies both (TH0) and (TH1). In topological spaces, (TH1) and (T0) are equivalent. There does not seem to exist any systematic exploration of topological notions of thinness. References Albuquerque J (1941) La notion de frontiere en topologie. Portug. Math. 2: Brissaud MM (1975) Les espaces prétopologiques. Comptes rendus de l Académie des sciences Paris Serie A 280: Čech E (1966) Topological Spaces. London: Wiley. Davey BA, Priestley HA (1990) Introduction to Lattice and Order. Cambridge UK: Cambridge Univ. Press. Day MM (1944) Convergence, Closure, and Neighborhoods. Duke Mathematical Journal 11: Dittrich P, Speroni di Fenizio P (2006) Chemical Organization Theory. Bulletin of Mathematical Biology In press. Preprint: q-bio.mn/ Eckhardt U, Latecki L (1994) Digital topology. Tech. Rep. 89 Hamburger Beiträge zur Angewandten Mathematik A. Gaal SA (1964) Point Set Topology. New York: Academic Press. Galton A (2000) Continuous Motion in Discrete Space. In: Principles of Knowledge Representation and Reasoning: Proceedings of the Seventh International Conference (KR2000) (Cohn AG, Giunchiglia F, Selman B, eds), San Francisco, CA: Morgan Kaufmann Publishers. Gastl GC, Hammer PC (1967) Extended topology. Neighborhoods and convergents. In: Proceedings of the Colloquium on Convexity 1965 (Fenchel W, ed), Copenhagen, DK: Københavns Universitets Matematiske Institut. Gniłka S (1994) On extended topologies. I: Closure operators. Annales de la Societé Polonaise de Mathematique, Ser. I, Commentat. Math. 34: Hammer PC (1955) General Topology, Symmetry, and Convexity. Transactions of the Wisconsin Academy Sciences, Arts, Letters 44: Hammer PC (1962) Extended Topology: Set-Valued Set Functions. Nieuw Archiv voor Wiskunde III 10: Hausdorff F (1935) Gestufte Räume. Fundamenta Mathematicae 25: Largeron C, Bonnevay S (2002) A pretopological approach for structural analysis. Information Sciences 144: LeBourgeois F, Bouayad M, Emptoz H (1999) Structure Relation between Classes for Supervised Learning Using Pretopology. In: Fifth International Conference on Document Analysis and Recognition Marchand-Maillet S, Sharaiha YM (1997) Discrete Convexity, Straightness, and the 16- Neighborhood. Computer Vision & Image Understanding 66: Biological Theory 1(3) 2006, Supplementary Electronic Materials 4

5 McKenzie R (1971) Cardinal Multiplication of structures with a reflexive multiplication. Fundamenta Mathematicae 70: Pfaltz J (1996) Closure Lattices. Discrete Mathematics 154: Smyth MB (1995) Semi-metric, closure spaces and digital topology. Theoretical Computer Science 151: Speroni di Fenizio P, Banzhaf W, Ziegler J (2002) Towards a theory of organizations. In: Proceedings of the Fourth German Workshop on Artificial Life (GWAL 00) (Lange H, ed),. Stadler BMR, Stadler PF (2003) Higher Separation Axioms in Generalized Closure Spaces. Commentationes Mathematicae Warszawa, Ser. I 43: Stadler BMR, Stadler PF, Wagner G, Fontana W (2001) The topology of the possible: Formal spaces underlying patterns of evolutionary change. Journal of Theoretical Biology 213: Steen LA, Seebach J (1970) Counterexamples in Topology. New York: Holt, Rinehart & Winston. Thampuran D (1971) Normal Neighborhood Spaces. Rendiconti del Seminario Matematico della Universita di Padova 45: Biological Theory 1(3) 2006, Supplementary Electronic Materials 5