Search results for "Topological manifold"
showing 8 items of 8 documents
On Słowikowski, Raíkov and De Wilde Closed Graph Theorems
1986
Publisher Summary This chapter focuses on the Slowikowski, Raikov and De Wilde closed graph theorems. The vector spaces used in the chapter, are defined over the field Ղ of real or complex numbers. The term, “space” means separated topological vector space, unless the contrary is specifically stated. If Ω is a non-empty open subset of the n -dimensional euclidean space, then the Schwartz space ҟ′(Ω) endowed with the strong topology belongs to this class. The chapter also studies the classes of spaces related with this conjecture. The class of Slowikowski spaces contains the F-spaces and it is stable with respect to the operations that include: countable topological direct sums, closed subsp…
On i-topological spaces: generalization of the concept of a topological space via ideals
2006
[EN] The aim of this paper is to generalize the structure of a topological space, preserving its certain topological properties. The main idea is to consider the union and intersection of sets modulo “small” sets which are defined via ideals. Developing the concept of an i-topological space and studying structures with compatible ideals, we are concerned to clarify the necessary and sufficient conditions for a new space to be homeomorphic, in some certain sense, to a topological space.
Noetherian type in topological products
2010
The cardinal invariant "Noetherian type" of a topological space $X$ (Nt(X)) was introduced by Peregudov in 1997 to deal with base properties that were studied by the Russian School as early as 1976. We study its behavior in products and box-products of topological spaces. We prove in Section 2: 1) There are spaces $X$ and $Y$ such that $Nt(X \times Y) < \min\{Nt(X), Nt(Y)\}$. 2) In several classes of compact spaces, the Noetherian type is preserved by the operations of forming a square and of passing to a dense subspace. The Noetherian type of the Cantor Cube of weight $\aleph_\omega$ with the countable box topology, $(2^{\aleph_\omega})_\delta$, is shown in Section 3 to be closely related …
ON TOPOLOGICAL SPACES WITH A UNIQUE QUASI-PROXIMITY
1994
Abstract Trying to solve the question of whether every T 1 topological space with a unique compatible quasi-proximity should be hereditarily compact, we show that it is true for product spaces as well as for locally hereditarily Lindelof spaces.
Volume-convergent sequences of Haken 3-manifolds
2003
Abstract Let M be a closed orientable 3-manifold and let Vol(M) denote its Gromov simplicial volume. This paper is devoted to the study of sequences of non-zero degree maps f i :M→N i to Haken manifolds. We prove that any sequence of Haken manifolds (Ni,fi), satisfying limi→∞deg(fi)×Vol(Ni)=Vol(M) is finite up to homeomorphism. As an application, we deduce from this fact that any closed orientable 3-manifold with zero Gromov simplicial volume and in particular any graph manifold dominates at most finitely many Haken 3-manifolds. To cite this article: P. Derbez, C. R. Acad. Sci. Paris, Ser. I 336 (2003).
A topological model for Oersted-Amp�re's law
1973
A geometrical description of Oersted-Ampere's law ∮H ds=(4π/c)I can be given in terms of an appropriate topological manifold. More precisely: It will be shown that Oersted-Ampere's law can be related to the topological invariantH 1(S 1), i.e. de Rham's first cohomology group on the differentiable manifoldS 1={(x,y) ∈ ℝ2∶x 2+y 2}
Localification of variable-basis topological systems
2011
The paper provides another approach to the notion of variable-basis topological system generalizing the fixed-basis concept of S. Vickers, considers functorial relationships between the categories of modified variable-basis topological systems and variable-basis fuzzy topological spaces in the sense of S.E. Rodabaugh and shows that the procedure of localification is possible in the new setting. Quaestiones Mathematicae 33(2010), 11–33
Countable connected spaces and bunches of arcs in R3
2006
Abstract We investigate the images (also called quotients) of countable connected bunches of arcs in R 3 , obtained by shrinking the arcs to points (see Section 2 for definitions of new terms). First, we give an intrinsic description of such images among T 1 -spaces: they are precisely countable and weakly first countable spaces. Moreover, an image is first countable if and only if it can be represented as a quotient of another bunch with its projection hereditarily quotient (Theorem 2.7). Applying this result we see, for instance, that two classical countable connected T 2 -spaces—the Bing space [R.H. Bing, A connected countable Hausdorff space, Proc. Amer. Math. Soc. 4 (1953) 474], and th…