Tychonoff Theorem Research Articles

COMPACTNESS DEGREE AND A GENERALIZATION OF THE TYCHONOFF THEOREM

The most interesting result of this paper is the following generalization of the Tychonoff theorem: CD(IIα Xα) = minα{CD(Xα)}.

Definitions of compact

Several definitions of “compact” for topological spaces have appeared in the literature (see [5]). We will consider the following:Definition. A topological space X is1. Compact(1) if every open cover of X has a finite subcover.2. Compact(2) if every infinite subset E of X has a complete accumulation point (i.e., a point x0 ∈ X such that for every neighborhood U of x0, |E ∩ U| = |E|).3. Compact(3) if there is a subbase S for the topology on X such that every cover of X by members of S has a finite subcover.4. Compact(4) if each nest of closed, nonempty sets has a nonempty intersection.5. Compact(5) if every family of closed sets in X which has the finite intersection property (every finite subfamily has a nonempty intersection) has a nonempty intersection.6. Compact(6) if each net in X has a cluster point.7. Compact(7) if each net in X has a convergent subnet.This work was motivated primarily by consideration of various proofs that the Tychonoff theorem, T (“the product of compact topological spaces is compact”) is equivalent to the Axiom of Choice, AC. Tychonoff's original proof that AC implies T used Definition 2 [13]. Other proofs have used Definitions 3 and 5; see [5]. The proof by Kelley that T implies AC uses Definition 5 [6].

Topological structures in fuzzy lattices

In his classical paper [2 1 ] Zadeh introduced the fundamental concept of fuzzy sets. After that appeared many results about topological structures on fuzzy sets: fuzzy topologies [ 1, 3,4, 6, 11-13, 16-201, fuzzy proximities [7-91, fuzzy uniformities [5, 11, 141, and, recently, a synthesis of these theories-the theory of fuzzy syntopogenous structures [lo]. We remark that in most cases the definitions and the results of papers mentioned above are valid in any completely distributive lattice. A large number of results concerning topological structures in lattices can be found in Nobeling’s book [ 151. So, the definition of fuzzy topology, as well as the definitions of neighbourhoods, interior, and compactness of [ 1 ] follows from [ 15, pp. 42, 44-46, resp. 951, and Theorems 2.2, 2.3, 4.2, and 5.1 follow from 7.19, 7.17, 10.3-10.6, and 12.3. Theorems 3.1 and 3.2 of [ 181 are consequences of [ 15, Theorems 12.2, 12.111. The definition of product fuzzy topology, and Theorem 3.1 of [ 191 correspond to [ 15, p. 13 1, and Theorems 17.1, 17.21. The fuzzy variant of the Tychonoff theorem (see 13, 2.1; 19, Theorem 3.41) follows from [lS, 17.71. The definition of fuzzy uniformity (see [5]) is equivalent to the definition from [ 15, p. 1691 (in the case of the complemented lattices). For the construction of the topology induced by uniformity and the definition of uniform continuity of mappings see [15, 19.1; p. 1851.

Extensions and new observations of Tychonoff, Stone-Weierstrass theorems, compactifications and the realcompactification

An extension of the Tychonoff theorem is obtained in characterizing a compact space by the nets and the images induced by any family of continuous functions on it. The idea of this extension is applied to get a new process and new observations of compactifications and the realcompactification. Finally, a sufficient and necessary condition of a vector sublattice or a subalgebra of C ∗( X) to be dense in ( C ∗( X),∥·∥) is provided in terms of the nets in X induced by C ∗( X), where C ∗( X) is the space of all bounded real continuous functions on a topological space X with pointwise ordering, and ∥·∥ is the supremum norm.

Tychonoff's theorem without the axiom of choice

Tychonoff's theorem without the axiom of choice

Probabilistisch kompakte L-unscharfe Mengen

In the case of completely distributive lattices (L,≤) we establish a probabilistic version of Alexander's subbase lemma and of Tychonoff's theorem for L-fuzzy sets. As an application we obtain that probabilistic topologies induced by compact ordinary topologies are also compact; i.e. ordinary compactness is consistent with probabilistic compactness. Regarding the validity of these results a counterexample shows that the complete distributivity of (L,≤) cannot be replaced by a weaker distributivity condition.

A comparison of different compactness notions in fuzzy topological spaces

In this paper we study the different kinds of compactness notions that have been introduced up to this time. We restrict ourselves to fuzzy topological spaces as defined in [4]. In Section 1 we give some alternative characterizations. For a good definition of fuzzy compactness we will demand that in the special case of topological spaces it coincides with the usual notion of compactness. In Section 2 we show which compactness notions are good extensions and which are not. Moreover, we want the fundamental property of compactness in topological spaces, namely, the Tychonoff theorem on products, to be fulfilled in the more general setting of fuzzy topological spaces. In Section 3 we see for which notions there is a .product theorem. In Section 4 we study the implications that exist between the different notions, and in Section 5 we give some concluding remarks. 1.

Products of topological spaces

The main purpose of this paper is to unify a number of theorems in topology whose conclusions state that a product of topological spaces has a compactness-like property.Three such theorems are (1) the Tychonoff theorem: Every product of compact spaces is compact, (2) the theorem of C.T. Scarborough and A.H. Stone: Every product of at most N 1 sequentially compact spaces is countably compact, and (3) the theorem of N. Noble: A countable product of Lindelöf P-spaces is Lindelöf.

Compactness in fuzzy topological spaces

Since earlier approaches to compactness in fuzzy spaces have serious limitations, we propose a new definition of fuzzy space compactness. In doing so, we observe that it is possible to have degrees of compactness, which we call α-compactness (α a member of a designated lattice). We obtain a Tychonoff Theorem for an arbitrary product of α-compact fuzzy spaces and a 1-point compactification. We prove that the fuzzy unit interval is α-compact.

Initial and final fuzzy topologies and the fuzzy Tychonoff theorem

In this paper we introduce the notions of initial and final fuzzy topologies. In Theorems I .4 and 2.4, we show that from a categorical point of view they are the right concepts to generalize the topological ones. In Theorem 1.6, we show that with our notion of fuzzy compactness the Tychonoff product theorem is safe- guarded. We also show that this is not the case for weak fuzzy compactness. For notions and results used but not defined or shown in this paper, we refer the reader to [lo]. 1.

A note on the topolgy of 𝐶-convergence in hyperspaces

In this note we generalize and partially correct a recent Tychonoff theorem for hyperspaces of F. A. Chimenti [1].

Generalizations of boundedness, compactness and the tychonoff theorem

A restrictedness on a topological space X is a collection K of subsets of X, the elements of which are said to be restricted. X is locally restricted if every point has a restricted neighborhood. A type of restrictedness is a collection T of restrictednesses (not necessarily all on the same space); T ( X) will denote the collection of members of T which are restrictednesses on the space X. X is called T -compact if for every R ϵ T ( X), X is restricted if it is locally restricted. The Tychonoff Theorem is generalized in a form which gives a necessary and sufficient condition for a product of spaces to be T -compact for any given type T . Some applications are given, and several noteworthy types of restrictedness are introduced, including ones which display compactness, connectedness and H-closedness as special cases of a more general concept.

A General Tychonoff Theorem for Multifunction

Let {Xa}a∈A be a non-empty family of compact spaces and let F be a subset of the m-product P{Xa:a∈A}. The Tychonoff problem is the determination of the sets F which are pointwise compact. It is known that P{Xa: a ∈ A} is pointwise compact [5, p. 400]. Also the set F of all point-closed multifunctions of P{Xa:a∈ A) is pointwise compact.

The fuzzy tychonoff theorem

Much of topology can be done in a setting where open sets have “fuzzy boundaries.” To render this precise, the paper first describes cl ∞ -monoids, which are used to measure the degree of membership of points in sets. Then L- or “fuzzy” sets are defined, and suitable collections of these are called L-topological spaces. A number of examples and results for such spaces are given. Perhaps most interesting is a version of the Tychonoff theorem which gives necessary and sufficient conditions on L for all collections with given cardinality of compact L-spaces to have compact product.

Continuity of higher derivations

Let A and B be two algebras over the complex field C. A sequence {Fn}0ggra (resp.{F}0gn A is equicontinuous. A is termed functionally continuous if every multiplicative linear functional on A is continuous. It is an open question whether A is necessarily functionally continuous, even in the locally w-convex case, though it is known that some metrizability condition is necessary. In the case A locally convex with identity having a neighbourhood of in- vertible elements (so O^ is compact by Tychonoff's theorem), Q)A point separating and B=Â, Johnson (7) has shown that any derivation from A into B is continuous. Indeed, for A a Banach algebra he obtained results for higher derivations (Dkik\)k3.0 on A, where D is a derivation on A. In this note we show how the argument of (7) works in the more general situation to give results on the continuity of higher derivations. Our results contain those of the relevant sections of (3) and (12) where regular semi- simple £-algebras are considered. After completing an earlier draft of this

Tychonoff's Theorem for Hyperspaces

Tychonoff's Theorem for Hyperspaces

Some combinatorial theorems equivalent to the prime ideal theorem

Some useful combinatorial selection lemmas are shown to be directly equivalent to the prime ideal theorem for boolean algebras. 1. The theorems we shall consider are intimately related to R. Rado's selection lemma (Theorem 2, below) which first appeared in [10] and subsequently has found wide application (see [1], [3], [4], [12]). Our main theorem is Theorem 1 which we use to derive other forms of Rado's lemma and to prove A. Robinson's valuation lemma which was shown by Robinson in [9] to be a fundamental result in model theory. We also show that Theorem 1 and some of the theorems we derive from it are equivalent to the prime ideal theorem for boolean algebras and thus we have some useful abstract versions of this theorem, versions divorced from any particular algebraic structure. Whether the original lemma of Rado is as strong as the prime ideal theorem appears to be an open question; note, however, that E. S. Wolk [12] has shown that Rado's lemma plus the axiom of choice for families of finite sets implies the Tychonoff theorem for finite spaces, which, in turn, implies the prime ideal theorem (see [8]). We conjecture that Rado's lemma is strictly weaker than the prime ideal theorem. 2. Preliminaries. In this paper we shall work, informally, within the framework of Zermelo-Fraenkel set theory (ZF). All uses of the axiom of choice will be explicitly noted. Given a set I, by a partial function on I, we mean a function whose domain is a subset of L A partial valuation on I is a partial function on I whose range is included in {0, 1}. As in ZF we consider functions to be special sets of ordered pairs and we even allow 0, the empty function (the empty set of ordered pairs). Iff is a partial function on I we write D(f) for its domain, and if Uc I we writef L U for the restriction off to UriD(f) Presented to the Society, August 30, 1972; received by the editors November 28,

A general compactness theorem

It is the purpose of this paper to prove a general compactness theorem and deduce from it, as corollaries, the Tychonoff theorem on the product of compact spaces and Ascoli-type theorems giving sufficient conditions for function spaces with the topology of uniform convergence on compact sets to be compact.

A General Selection Principle, with Applications in Analysis and Algebra

The General Selection Principle referred to in the title is really Tychonoff's theorem on products of compact spaces, but in a somewhat disguised form (c.f. Theorem 2.1, below). It is believed that this form is one which lends itself very well to many applications. More specifically, one of the immediate corollaries of the main theorem is a theorem due to Rado (c.f. Cor 2.1.1.), which has been used by Erdos and de Bruijn [2], Luxemburg [7], and the author [9] to give simple proofs of a variety of results.

Tychonoff's Theorem for the Space of Multifunctions

Tychonoff's Theorem for the Space of Multifunctions