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Topology in lattices /
Abstract
Thesis (Ph. D.)--Michigan State University of Agriculture and Applied Science. Dept. of Mathematics, 1953.
For a fixed set X, an arbitrary \textit{weight structure} can be interpreted as a distance assignment between
pairs of points on X. Restrictions (i.e. \textit{metric axioms}) on the
behaviour of any such d naturally arise, such as separation, triangle
inequality and symmetry. We present an order-theoretic investigation of various
collections of weight structures, as naturally occurring subsets of
satisfying certain metric axioms. Furthermore, we
exploit the categorical notion of adjunctions when investigating connections
between the above collections of weight structures. As a corollary, we present
several lattice-embeddability theorems on a well-known collection of weight
structures on X.
In this article we give a purely topological characterization for a topology J on a set X to be the order topology with respect to some linear order R on X, as follows. A topology J on a set X is an order topology is a T1-space and J is the least upper bound of two minimal T0-topologies [Theorem 1]. From this we deduce a purely topological description of the usual topology on the set of all real numbers. That is, a topological space (X, J) is homeomorphic to the reals with the usual topology is a connected, separable, T1-space, and J is the least upper bound of two noncompact minimal T0-topologies [Theorem 2].
The object is to determine what theorems for single-valued functions can be extended to which class of multi-valued functions. It is shown that an arc cannot be mapped onto a circle by a continuous, monotone multi-valued function when the image of each point is an arc. On the other hand, the arc can be mapped onto a nonlocally connected space by a monotone, continuous function such that the image of each point is an arc. Characterizations of nonalternating functions analogous to the results in the single-valued theory are obtained, and it is shown that an nonalternating semi-single-valued continuous function on a dendrite is monotone. An analog of the monotone light factorization theorem is obtained for semisingle- valued continuous functions. Some other results are: an open continuous function with finite images maps a regular curve onto a regular curve, and a continuous function with finite images maps a locally connected, compact space onto a locally connected compact space.
In this paper, the notion of an n-ordered set is introduced as a natural generalization of that of a totally ordered set (chain). Two axioms suffice to describe an w-order on a set, which induces three associated structures called respectively: the incidence, the convexity, and the topological structures generated by the order. Some properties of these structures are proved as they are needed for the final theorems. In particular, the existence of natural k-orders in the “flats” of an n-ordered set and the fact that (as it happens for chains) the topological structure is Hausdorff. The idea of Dedekind cut is extended to n-ordered sets and the notions of strong-completeness, completeness, and conditional completeness are introduced. It is shown that the Sn sphere is s-complete when considered as an n-ordered set. It is also proved that En, the n-dimensional euclidean space, fails to be s-complete or complete, but that it is conditionally complete. It is also proved that every s-complete set is compact in its order topology but that the converse is not true. These results generalize classical ones about the structure of chains and lattices.
Let G be a locally compact idempotent, commutative, topological semigroup (semi-lattice). Let M(G) denote its measure algebra, i.e., M(G) consists of all countably additive regular Borel-measures defined on G and has the usual Banach algebra structure: pointwise linear operations, convolution, and total variation norm. To understand the structure of such a convolution algebra one studies its maximal ideals, the nature of the Gelfand transform, the structure of the closed ideal and the related question of spectral synthesis, etc. In this paper G is the cartesian product of topological semigroups Gα of the following form: Gα is a linearly ordered set, locally compact in its order topology; multiplication in Ga is given by xy = max (x, y). The product semigroup is assumed locally compact in the product topology. The main theorem of this paper gives a representation of the space of maximal ideals ΔM(G), for a finite product, in terms of the dual semigroup Ĝ. The multiplicative linear functionals of M(G) are integrals of fixed semi-characters It is shown that this integral representation does not hold for infinite products because the semi-characters are usually not integrable.
It is shown that if (L, T) is a compact connected modular topological lattice of finite dimension under a topology T, then the topology T, the interval topology of L, the complete topology of L, and the order topology of L are all the same.
Important classes of topological spaces have topologies which are induced by a generating collection of closed subsets; typical examples are K-spaces, sequential spaces with unique sequential limits, and lattices with the Birkhoff interval topology. This paper proceeds by axiomatizing this construction −a set with a specified generating collection of closed subsets is called a “hypotopological space.”The Birkhoff interval topology is then studied in these terms. A natural embedding of hypotopological spaces in conditionally complete, atomic, distributive lattices with Birkhoff interval topology is derived. This embedding is used to show that lattices with Birkhoff interval topology have the same nontrivial subspace and product properties as K-spaces and sequential spaces. In particular, we answer in the negative a question first raised by Birkhoff, namely, whether the Birkhoff interval topology is preserved under the formation of the product of two lattices.
The intrinsic topology s≤of a chain (X, ≤) induces on any subchain Y⊂X the relative topology s≤Y. On the other hand, any such subchain Y is endowed with its own intrinsic topology s≤y. We establish several necessary and sufficient conditions under which both topologies coincide, by suitably weakening the properties of convexity (Lemma 2), order-density (Theorem 3) and subcompleteness (Theorem 4), respectively. Another necessary and sufficient condition for the equation s≤y = s≤yformulated in terms of cuts, is given in Theorem 2. Besides other related results, we find a purely order-theoretical characterization of those subchains which are compact (Lemma 1) or connected (Corollary 2), respectively, in the intrinsic topology of the entire chain. As a simple consequence of Theorem 4, we obtain the wellknown result that the intrinsic topology of a chain can be obtained by relativization from the intrinsic topology of the normal completion (Corollary 9). We conclude with several applications to the Euclidean topology on R. © 1979, University of California, Berkeley. All Rights Reserved.
We analyze the exploitation of an antibiotic in a market subject to open access on the part of antibiotic producers to the common pool of antibiotic efficacy. While the market equilibrium depends only on current levels of antibiotic efficacy and infection of the epidemiological system, the social optimum accounts for the dynamic externalities which relate those levels to the intertemporal use being made of the antibiotic. We show that depending on the parameters of the model, in particular the cost of production and the improvement in the recovery rate that results from antibiotic treatment, the positive steady-state level of antibiotic efficacy to which the system tends under open access can be lower or higher than the level which should prevail in the socially optimal steady state. In fact there are parameter configurations for which the steady states can be exactly the same. However, the paths leading to the steady state always differ.
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