Dugundji, "Topology", Allyn and Bacon, Boston 1966.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....The following notation is used throughout: denotes the set of natural numbers, denotes the set of real numbers and = 0, #) For standard topological notions such as a topology, the supremum topology i#I i of a family of topologies (T i ) i#I etc. we refer the reader to e.g. [Dug66]. If (X, 1 , 2 ) is a triple consisting of a set X and two topologies 1 and 2 on X, then the notation int A, where A X and i # 1, 2 , indicates the interior of the set A with respect to the topology i . A topology on a set X is T 0 if # #O # T O and y O ....

....a suitable condition, a sequential completion is adequate; that is whether a general completion based on filters or nets is replaceable by such a sequential completion. If this is the case, we say that the general completion is sequentially adequate (under the given condition) For instance, in [Dug66], it is shown that for a uniform space with a countable base, sequential completions are adequate. The question is also relevant from a computer science point of view, since the possibility to work with sequences rather than with nets or filters leads to a considerable simplification of the ....

J. Dugundji, Topology, Allyn and Bacon, Inc., Boston, 1966.

....similar statement for an lR valued , with j j 1, and analogous assumptions as in (4.6) 4.7) One now chooses for 0 and in the closed unit ball of lR ; x; e) 0 (x; e) 0 (x; e ; in place of (4. 8) and uses Brouwer s xed point theorem, cf. Dugundji [3], p. 341, to nd for 0 , satisfying the second equality of (4.13) This remark may be helpful if one wishes that the distribution of Theorem 4.1 accommodates a genuinely vector valued local drift. We now proceed with the proof of Theorem 4.1. We are reduced to checking the ....

J. Dugundji. Topology. Allyn and Bacon, Boston, 1966.

....these must be of infinite index in M . Let L i = #(M i ) So L i is a compact (in particular, closed) proper subgroup of L. It is of infinite index in L, so is nowhere dense. So we have that L is compact and the union of a countable chain of nowhere dense subsets. This contradicts Baire s Theorem ([3], XI Theorem 10.1, for example) Remarks 3.4. Thus the topology on a weakly profinite group M is determined by any countable chain of compact subgroups with union M . Lemma 3.1 says that any two such chains are cofinal in each other. Note also that we can define a weakly profinite group by a ....

James Dugundji, Topology, Allyn and Bacon, Boston, 1966.

....sizes, for example, have no significance. Although not quite as widely used as graphs, Euler circles, or Venn diagrams, are often used to represent logical propositions, color charts, etc. see Figure 2) The basic idea is to appeal to the two dimensional case of the Jordan curve theorem (e.g. [11, 30]) which estab lishes that simple closed curves partition the plane into disjoint inside and outside regions. A set is then represented by the inside of such a curve, 2 giving the topo logical notions of enclosure, exclusion, and intersection of the curves their obvious set theoretic meanings: ....

Dugundji, I. Topology.. Allyn and Bacon. Boston, Mass.. 1966.

....precisely the union of sets ( ffl balls ) of the form fy j d(x; y) fflg. A topological space is metrizable if there is a metric d that induces the topology. A topological metric is a binary function d such that the topology induced by d is metrizable. There is a theorem of Nagata and Smirnov [Dug66, pp. 193 195] that a topological space is metrizable if and only if it is regular and has a basis that can be decomposed into an at most countable collection of nbd finite families. The proof of the only if direction can be modified in an obvious manner to show that every topological space ....

J. Dugundji. Topology. Allyn and Bacon, Inc., Boston, 1966.

....long sequence of states leads. This is possible in a natural way if a topological structure on the state space of the labelled transition system is given. Topology is a field of mathematics in which general definitions of convergence and accumulation of sequences have been developed (see e.g. [7, 8]) In this paper we define the notion of a labelled topological transition system, i.e. a labelled transition system where the state space is structured using a topology. Then, we define topological simulation and topological bisimulation. These notions extend the traditional ones by considering ....

....this was, to our knowledge, never used with respect to bisimulation equivalence. Note that giving a metric on a set is only one way of inducing a topology. Alternatively, for example, a complete partial order gives rise to a topology as well [8, 13] The following definitions are taken from [7]. Definition 6 (Topology) Let X be a set, then T C 2 x is a topology on X if and only if 0 T, X T, every finite intersection of elements of T is again an element of T, and every arbitrary union of elements of T is again an element of T. The elements of T are called open sets. An open set U T ....

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J. Dugundji. Topology. Allyn and Bacon, Inc., Boston, 1966.

....then (UX, is Lawson compact, and hence a coherent domain. iii) If X is compact and zero dimensional then UX is w algebraic, i.e. it is a Scott domain. Proof (i) Since X is a second countable, locally compact Hausdorff space, it has a countable basis consisting of relatively compact open sets [Dugundji, 1966, page 238] Let 5 be the set of finite unions of closures of this basis. Then 5 is a countable subset of UX. For each (7 C UX, we let W( 7) B C 5 lB (7 . As in the case of W( 7) in the proof of Proposition 3.3, it follows that W( 7) is directed with lub (7. Therefore, 5 is a countable ....

....valuation on UX extends uniquely to a finite regular measure on UX equipped with its Lawson topology. Assume for the rest of this section that X is a second countable locally compact Hausdorff space. Since every locally compact Hausdorff space is regular, and since by Urysohn s Theorem [Dugundji, 1966, page 195, Corollary] every second countable regular Hausdorff space is metrizable, we can assume that X is in fact a metric space. We will now examine the relationship between the Borel subsets of X and those of UX. Proposition 5.9 Let a G X be an open subset of X. There exists a countable ....

J. Dugundji. Topology. Allyn and Bacon, 1966.

....if it is continuous at some point. Observe the analogy to homomorphisms of topological groups. Let us fix some notation. For the basic concepts of plane projective geometry, the reader is referred to Pickert [13] or Hughes Piper [12] for topological notions see e.g. Bourbaki [2] 3] Dugundji [5], Engelking [6] Four points of a projective plane are called a quadrangle if any three of them are not collinear. A topological projective plane Pi = P; L; P ; L ) is a projective plane (P; L) with neither indiscrete nor discrete topologies P on P and L on L, respectively, such that ....

....theorem is well known, cf. Arens Dugundji [1] Poppe [14] Theorem (2.34) 3) The second part is probably known, too, though it seems not to be mentioned in the literature. We omit its proof which uses standard arguments from the theory of uniform spaces. For a related result compare also Dugundji [5], Chapter XII, 7.5. Note that if in part (2) additionally, X were Hausdorff and f i were continuous (all i 2 I) 2) would be a consequence of (1) see, as mentioned above, Bourbaki [3] X.3.4, Theorem 2) 2.2) Theorem: Let (X; and (Y; AE) be topological spaces, f i ) i2I a net in Y , and ....

Dugundji, J.: Topology. Allyn and Bacon, Boston 1966

....the 4 corner points in the theorem proof. Whether i j may be considered a good estimate for d(C(i) C(j) m depends on the existence of a positive constant lower bound on L 2 (C) The answer to this is negative, relying on the following discrete analog of the classic topological theorem ([2], Chap. 5, Theorem 2.3) that no mapping f : 0, 1] # [0, 1] m (m 1) can be continuous and also possess a continuous inverse. Theorem 2 If C is a discrete m dimensional space filling curve on [N ] m , then L 2 (C) O(N 1 m ) Proof : Choose a segment S of C of length at least 1 4 N ....

J. Dugundji. Topology. Allyn and Bacon, Inc., Boston, 1966.

....preliminaries We assume a basic understanding of elementary metric topology and basic measure theory, in particular the notions of complete metric space, closed and compact subset, and algebra and Borel probability measure. We refer to [BV96] or standard textbooks on general topology such as [Dug76,Eng89] for more details on the former topic, and to [Rud66,Hal74] for more information on the latter. A metric space X is said to be an ultrametric space if, for all x; y; z 2 X, it holds that d(x; z) maxfd(x; y) d(y; z) g. We use B (x) with 0 and some element x in a metric space (X; d) to ....

J. Dugundji. Topology. Allyn and Bacon, 1976.

....sets in X. For every A; B 2 C(X) de ne the generalized Hausdor 22 distance by H(A; B) max max a2A min b2B d(a; b) max b2B min a2A d(a; b) It is easy to check that (C(X) H) becomes a metric space. In fact, we have the following general facts: Proposition 3 ([26, 27]) Let (X; d) be a metric space. 1) If (X; d) is compact, then (C(X) H) is a compact metric space. 2) If (X; d) is complete, then (C(X) H) is a complete metric space. Thus, in particular, the space (C n ; H) is a complete metric space for every n = 1; 2; Now we show the completeness ....

J. Dugundji, Topology, Allyn and Bacon, Inc., 1966.

....between p and q . This distance function is obviously symmetric. In this section we define the convex polygon offset distance function and show that in general it does not fulfill the triangle inequality. 2.1. Metrics and Convex Distance Functions A metric on the plane (see [KW, p. 16] and [Du]) is a mapping D: R 2 R 2 # R#0 such that for any points p, q, r # R 2 the following conditions are fulfilled: 1. D(p, q) 0 if and only if p = q (identity) 2. D(p, q) D(q, p) symmetry) and 3. D(p, r) # D(p, q) D(q, r) triangle inequality) For every convex polygon ....

J. Dugundji, Topology, Allyn and Bacon, Boston, MA, 1970.

....Complete Metric Spaces Let CMS be the category with complete metric spaces (D; dD ) as objects and nonexpansive (non distance increasing) functions as arrows. That is, functions f : D E such that, for all x; y 2 D, 29 dE (f(x) f(y) dD (x; y) For basic facts on metric spaces see, e.g. [Dug66]. For any two complete metric spaces D and E, the set of arrows between D and E, hom(D; E) j ff : D E j f is non expansiveg is itself a complete metric space, with metric, for all f; g 2 hom(D; E) d(f; g) j sup x2D fdE (f(x) g(x) g: In analogy to the so called order enriched (or O ) ....

J. Dugundji. Topology. Allyn and Bacon, inc., 1966.

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Dugundji, "Topology", Allyn and Bacon, Boston 1966.

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Dugundi, J.: Topology. Allyn and Bacon (1966)

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J. Dugundji, Topology, Allyn and Bacon, Boston, 1966.

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J. Dugundji, Topology, Allyn and Bacon, Boston, 1966.

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Dugundji, J., Topology, Allyn and Bacon, (1966).

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J. Dugundji, Topology, Allyn and Bacon (1966).

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J. Dugundji, Topology, Allyn and Bacon, Boston, 1966.

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J. Dugundji, Topology, Allyn and Bacon, Boston, 1966.

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J. Dugundji. Topology, pages 83--85. Allyn and Bacon, Inc., Boston, 1966.

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Dugundji, J. Topology. Allyn and Bacon, Boston, 1966. 39

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Dugundji, J., Topology, Allyn and Bacon, Boston, 1976.

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J. Dugundji. Topology. Allyn and Bacon, 1976.

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