R. Engelking, General Topology, Heldermann Verlag, Berlin, 1989.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....Thus, the topological structure is uniformly de ned just as a metric space, but, it is de ned not through a metric function but through a lter of open coverings (or relations) There are two di erent formalizations of uniform spaces. One is based on entourages of the diagonal [Bou65] Jam90] [Eng89], and the other is based on uniform coverings [Tuk40] Isb64] This paper uses the latter one, because it ts very well with our domain theoretical development. In this paper, we introduce the notion of a uniform domain, which is an algebraic domain with some uniformity condition. In a uniform ....

Ryszard Engelking. General Topology. Heldermann Verlag, Berlin, 1989. 12

....identity on X . When X is a subspace of Y , we say that X is a retract of Y if r and the embedding of X in Y form a retract. In this paper, we say that a topological space is compact when each open cover has a nite subcover and we do not assume the Hausdor property. See, for example, Smy92] and [Eng89] for topological notions. 4 Filter and Filter base A lter in a topological space X is a non empty family F of subsets of X which satis es the following conditions: 1. if A 2 F and A B, then B 2 F , 2. if A 1 2 F and A 2 2 F , then A 1 A 2 2 F , 3. 62 F . A lter base in X is a ....

....x. A point x is called a cluster point of a lter F (or a lter base B) if x belongs to the closure of every element of F (or B) We say that a lter (or a lter base) F 1 re nes F 2 if F 1 F 2 . We say that a lter (or a lter base) F is in nite when F is an in nite family. See, for example, [Eng89] for the notion of lters. The Real Line A dyadic number is a rational number of the form m 2 for integers m and n. We write I for the unit closed interval [0; 1] 3 Coherent domains We use domains to represent topological structures; we embed a topological space X in L(D) and consider K(D) ....

Ryszard Engelking. General Topology. Heldermann Verlag, Berlin, 1989. 26

....sets j every open cover of f (V ) has a nite subcover of Ug; where U and V range over open sets of X and Y respectively. In applications of function spaces Y to analysis, the spaces X and Y are usually Hausdor , and this is the level of generality often considered in books on topology [3, 5, 14]. In this case, the exponential topology coincides with the more familiar compact open topology. Moreover, for Hausdor spaces, localcompactness is the same as core compactness. However, in applications of topology to algebra via Stone duality [11] and to the theory of computation [1, 17, 19, 20, ....

R. Engelking. General topology. Heldermann Verlag, Berlin, second edition, 1989.

.... of non compact coalgebras are given by the carriers Z b and (and inheriting the structure from #) X 1 and consider the nal coalgebra (Z, #) with Z = Then Z is compact in the Cantor space topology (since the limit of compact Hausdor spaces is compact Hausdor , see [5], 3.2.13) and b # is not compact. The topology on Z is as follows. A subset of Z is open i it is a subset of b # or of the form V ( a, b # ) for some V b # . In particular, every open cover of also covers b # . Another example is given by TX = X) for which the ....

....# 1 # = ## # 1 # = id T # 1 . Let p i = id T # 1 . It was shown in Corollary 4. 5 that (T 1, i) is nal in Beh# (T ) It is compact since T 1 is the limit of compact Hausdor spaces and the induced topology on a limit of compact Hausdor spaces is compact Hausdor (see [5] 3.2.13) Remark 7.5. An inspection of the proof shows that also holds for logics of rank #. Moreover, for , we can weaken Condition 7.2 and only require that elements of T 1, n #, are closed wrt. # n . On the other hand, does not hold for logics of rank # as can be seen ....

Ryszard Engelking. General Topology. Heldermann Verlag, 1989.

....carriers Z b and s a : s a,b (and inheriting the structure from ( Example 6.13. Let TX a, b x X I and consider the final coalgebra (Z, with Z a, b U a, b . Then Z is compact in the Cantor space topology (since the limit of compact Hausdorff spaces is compact Hausdorff, see [5], 3.2.13) and a, b is not compact. The topology on Z is as follows. A subset of Z is open iff it is a subset of a, b or of the form V. a, b a, b ) for some V C a, b . In particular, every open cover of a, b also covers a, b . Another example is given by TX 7 (X) for which the ....

Ryszard Engelking. General Topology. Heldermann Verlag, 1989.

....carriers Z n fb g and fs a g (and inheriting the structure from ) Example 5.9 Let TX = fa; bg X 1 and consider the nal coalgebra (Z; with Z = fa; bg [ fa; bg . Then Z is compact in the Cantor space topology (since the limit of compact Hausdor spaces is compact Hausdor , see [4], 3.2.13) and fa; bg is not compact. The topology on Z is as follows. A subset of Z is open i it is a subset of fa; bg or of the form V (fa; bg fa; bg for some V fa; bg . In particular, every open cover of fa; bg also covers fa; bg . Example 5.10 Let T = P . Then the ....

....of Beh (T ) is compact in the induced topology. Proof Let p 1 . It was shown in Corollary 4. 5 that (T 1; i) is nal in Beh (T ) It is compact since T 1 is the limit of compact Hausdor spaces and the induced topology on a limit of compact Hausdor spaces is compact Hausdor (see [4] 3.2.13) 2 We can summarise: Theorem 6.6 Let T map nite sets to nite sets. Then Beh (T ) has a nal object that is compact in the Cantor space topology i T weakly preserves the limit of (T 1) n2N . 7 Conclusions and Related Work We have studied de nability and compactness for nitary ....

Ryszard Engelking. General Topology. Heldermann Verlag, 1989.

....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 ....

....a metric to define structure on the labels. Furthermore, 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 ....

R. Engelking. General Topology. Heldermann Verlag, 1989.

....carriers Z n fb g and fs a g (and inheriting the structure from ) Example 5.9 Let TX = fa; bg X 1 and consider the nal coalgebra (Z; with Z = fa; bg [ fa; bg . Then Z is compact in the Cantor space topology (since the limit of compact Hausdor spaces is compact Hausdor , see [4], 3.2.13) and fa; bg is not compact. The topology on Z is as follows. A subset of Z is open i it is a subset of fa; bg or of the form V (fa; bg fa; bg for some V fa; bg . In particular, every open cover of fa; bg also covers fa; bg Example 5.10 Let T = P . Then the ....

.... nal model of Log(T ) is compact in the induced topology. Proof Let p 1 . It was shown in Remark 4. 5 that (T 1; i) is nal in Log(T ) It is compact since T 1 is the limit of compact Hausdor spaces and the induced topology on a limit of compact Hausdor spaces is compact Hausdor (see [4] 3.2.13) 2 We can summarise: Theorem 6.6 Let T map nite sets to nite sets. Then Log(T ) has a nal object that is compact in the Cantor space topology i T weakly preserves the 17 limit of (T 7 Conclusions and Related Work We have studied de nability and compactness for nitary ....

Ryszard Engelking. General Topology. Heldermann Verlag, 1989.

....above claims are false. The aim of this note is among other things to show that for locally connected Polish spaces the gap between Dimensionsgrad and dimension can be arbitrarily large. 2. Preliminaries We start with a few simple and well known results. For details and undefined notions, see [3, 4]. Proposition 2.1. Let X be a space having the following property: every point has arbitrarily small connected neighborhoods (not necessarily open) Then X is locally connected. The next statement is a simple corollary to Proposition 2.1. Proposition 2.2. Let f : X # Y be a closed surjective ....

R. Engelking, General topology, Heldermann Verlag, Berlin, 1989.

....of [ Gamma1; 1] II . In any case, f defines another topology f on X , which is weaker (coarser) than the original one: oe f . Since f is metrisable, it is Hausdorff (and so is Hausdorff as well) Moreover, by the well known minimal property of compact topologies (see e.g. [4], Corollary 3.1.14, p. 126) both topologies coincide on compact sets, hence compact sets are metrisable and f is a homeomorphism, if restricted to each compact subset K ae X . In particular, f is a measurable isomorphism, if restricted to each oe compact subspace of (X ; Condition ....

Engelking, R., General Topology, Heldermann Verlag, Berlin 1989.

....rest of the IIMs. Clearly, families M j are disjoint and all of them are required to build a team identifying W . X 4 Some Topology Next, we present the notions from Topology that will be used in our proof below. Also presented are some examples that will play an integral role in the proof. See [Eng89, Kel57] for a further explanation of the concepts in this section. Definition 1 Let X be nonempty set and C a family of subsets (they will be called closed sets) of X satisfying the following conditions: 1. X 2 C and ; 2 C (The whole X and ; are closed) 2. If F 1 ; F 2 2 C, then F 1 [F 2 2 C (Union ....

R. Engelking. General Topology. Heldermann Verlag, Berlin, 1989.

....functions, He is also assumed to be familiar with Tarski s lemma, describing the set of prefixed points of continuous functions of complete lattices, and Banach s theorem, stating the existence of a unique fixed point of contractions in complete metric spaces. He is referred to [Ll87] and [En89], when need be. Furthermore, lack of space prevents us from describing all the metrics used in this paper. We will however employ the classical ones and refer to [B91] for such a description. 4 Semantic translation As pointed out in Section 1, the operator , acting in goals, can be simulated ....

Engelking R., General Topology, Heldermann Verlag, 1989.

....functions, He is also assumed to be familiar with Tarski s lemma, describing the set of prefixed points of continuous functions of complete lattices, and Banach s theorem, stating the existence of a unique fixed point of contractions in complete metric spaces. He is referred to [17] and [10], when need be. Furthermore, lack of space prevents us from describing all the metrics used in this paper. We will however employ the classical ones and refer to [7] for such a description. As a matter of notation, P nco (E) is subsequently used to denote the set of non empty and compact sets with ....

R. Engelking. General Topology. Heldermann Verlag, 1989.

....introduced into semantics in papers of Nivat [40] and de Bakker and Zucker [16] In this section we present two main theorems, Banach s fixed point theorem and Michael s theorem, which will be used frequently in the rest of this paper. For further reference considering metric spaces we suggest [26]. First we show how we can compose metric spaces. In the following definition we give some possible compositions, which will be used in the rest of this paper. Definition 2.1 Let (X; dX ) X 1 ; dX 1 ) and (X 2 ; dX 2 ) be metric spaces, where dX : X Theta X [0; 1] dX 1 : X 1 Theta X 1 ....

R. Engelking. General Topology. Heldermann Verlag (1989).

....LEX (U ) so T = LEX (U) is EX maximal. X For any set W F denote FEX (LEX (W ) by W . Let U; V be arbitrary classes of recursive functions. We summarize some of the previous results: ffl U U , ffl U V ) U V , ffl U = U . But U [ V 6= U [ V , therefore U U is not a closure operator [Eng89] To see that U [ V 6= U [ V , consider U; V 2 EX such that U [ V 62 EX, as in Theorem 10. Then U [ V = F , but U [ V = U [ V 6= F . Therefore the relation EX ae [1; 2]EX upsets this first attempt to introduce useful dual notions. Another approach will be given in the next section. 6 ....

R. Engelking. General Topology. Heldermann Verlag, Berlin, 1989.

....functions, He is also assumed to be familiar with Tarski s lemma, describing the set of prefixed points of continuous functions of complete lattices, and Banach s theorem, stating the existence of a unique fixed point of contractions in complete metric spaces. He is referred to [15] and [9], when need be. Furthermore, lack of space prevents us from describing all the metrics used in this paper. We will however employ the classical ones and refer to [5] for such a description. 5 Semantic translation As pointed out in section 1, synchronization can be specified in two places: in ....

R. Engelking. General Topology. Heldermann Verlag, 1989.

....computes forever writing p on the output tape. Note, that the type (finite or infinite) for each input and output tape must be defined in advance for each TT machine M . The function computed by a TT machine M is denoted by f M . TTE uses some basic notations and facts from topology (see e.g. [4] or any other textbook) We shall consider the discrete topology d : fA : A Sigma g on Sigma and the Cantor topology C : fA Sigma : A Sigma g on Sigma . The set fx Sigma : x 2 Sigma g is a base of C . As a fundamental property, every function computed by a ....

....that ] is not a symbol in w for w 2 dom( Every finite prefix of a ffi name p of x contains only finitely many atomic properties of x which approximate x. Mathematically, this kind of approximation is described by the topology oe . Since oe identifies points, it is a T 0 topology [4]. Computability on oe and via ffi on M are fixed by the notation , which expresses how atomic properties can be handled concretely. Thus for any information structure (M; oe; oe characterizes approximation and computability on M . For topological spaces (X; and (X 0 ; 0 ) a ....

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R. Engelking General Topology, Heldermann Verlag, Berlin, 1989.

....naturally leads to a distance between them, as formalized in example 2, and therefore to metric spaces. The reader is assumed to be familiar with metric spaces as well as with their related notions of convergent sequences, closed and compact subsets, completeness, and is referred to [9], when need be. For the sake of completeness, we just specify hereafter some practical language misuses, describe examples of metric spaces employed subsequently and recall the notion of contraction and its very useful property (due to S. Banach) of having one and only one fixed point in complete ....

R. Engelking. General Topology. Heldermann Verlag, 1989.

....earlier result on metrizable spaces, and by pulling back the obtained coloring, one obtains a coloring of the original homeomorphism. The aim of this note is to present a self contained proof of Theorem 1.1, using standard facts from dimension theory only. For all undefined notions, see Engelking [4, 5]. 1991 Mathematics Subject Classification. Primary: 54F45, Secondary: 54C05. Key words and phrases. color of a map, paracompact, n dimensional. 2 JAN van MILL 2. From finite to n 3 Let E be a collection of sets. As usual, we say that the order of E is less than or equal to n, abbreviated ....

R. Engelking, General topology, Heldermann Verlag, Berlin, 1989.

....( of Example 2.3(d) Prove that r is a semiprime ideal if and only if a 2 r implies that a 2 r. 3 A Biased Look at Topology. The references for basic topology are almost too many to count. Since the references to topology will be quite general I shall appeal to my favorites: Wi70] and [En89]. Later on, and in special circumstances, it will also be useful to refer to [GJ76] and [W74] For the record, we begin with a de nition of a topology. De nition 3.1. a) Let X be a set (possibly empty) A topology on X is a collection of subsets containing X and the empty set such that is ....

R. Engelking, General Topology. Heldermann Verlag (1989), Berlin.

....then is metrizable. Proof. For a metric space (X; d) the locally compactness and separability is equivalent to hemicompactness, that is the existence of an increasing sequence fK n g n of compact sets such that for every compact K 2 K(X) there is n 2 Z with K ae K n . X is locally compact [15], so by Proposition 4.3 the family fUK;ffl : K 2 K(X) ffl 0g is a base for a uniformity U on G. It is easy to verify that the family fU Kn ; 1 n : n 2 Z g is a countable base of U . Proposition 4.5. If (X; d) is a locally compact and separable metric space, then (G; is a Polish ....

R. Engelking: General Topology, Revised Edition, Heldermann Verlag, Berlin, 1989.

....d and ae such that B(CL(X) H d ) 6= B(CL(X) H ae ) if and only if X is not compact. Proof. If X is compact, then X has a unique compatible uniformity, and so every admissible metric gives the same Hausdorff metric topology. Conversely, if X is noncompact, then X admits an unbounded metric d [20]. But X can also be embedded into a countable product of intervals and thus also admits a totally bounded metric ae. By Theorem 3.1, the Borel fields for the induced Hausdorff metrics diverge. We next show that the Borel field of the Hausdorff metric topology agrees with the Effros sigma algebra ....

....topology. Let fx n : n 2 INg be a countable dense subset of X, and let be the topology on CL(X) generated by fd(x n ; Delta) n 2 INg [ fD d (B; Delta) B 2 Dg [ fe d (B; Delta) B 2 Dg; viewed as a family of functions on CL(X) As a weak topology, is automatically completely regular [20]. Since the family fd(x n ; Delta) n 2 INg separates points in the hyperspace, is Hausdorff. Since the family of generating functionals is countable, the topology is second countable. Thus, by the Urysohn metrization theorem, the topology is second countable and metrizable. Since d(x n ; ....

R. Engelking: General topology, Heldermann Verlag, Berlin 1989.

....in M and which we will need in the proof is in fact already in M. Moreover, most of our results will use elementary submodels of size 2 0 , it is worth noting that many of these results (and proofs) generalize to higher cardinals. For notation and terminology we refer the reader to [Ho] and [E]. For other applications of elementary submodels the reader is referred to [Do1] Do2] W1] W2] and [FW] 2. The results First we state some results which will allow us to simplify a certain number of proofs. Lemma 8. Let X be a T 1 space and let M be an elementary submodel which reflects ....

R. Engelking, General Topology, Heldermann Verlag, Berlin 1989.

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R. Engelking, General Topology, Heldermann Verlag, Berlin, 1989.

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