Topologies generated by ideals

Carlos Uzcátegui

Displaying similar documents to “Topologies generated by ideals”

Strong topologies on vector-valued function spaces

Marian Nowak (2000)

Czechoslovak Mathematical Journal

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Let $(X, ∥ \cdot ∥_{X})$ be a real Banach space and let $E$ be an ideal of $L^{0}$ over a $σ$ -finite measure space $(Ø, Σ, μ)$ . Let $(X)$ be the space of all strongly $Σ$ -measurable functions $f Ø \to X$ such that the scalar function $\tilde{f}$ , defined by $\tilde{f} (ø) = {∥ f (ø) ∥}_{X}$ for $ø \in Ø$ , belongs to $E$ . The paper deals with strong topologies on $E (X)$ . In particular, the strong topology $β (E (X), E {(X)}_{n}^{\sim})$ ( $E {(X)}_{n}^{\sim} =$ the order continuous dual of $E (X)$ ) is examined. We generalize earlier results of [PC] and [FPS] concerning the strong topologies.

The ideal (a) is not $G_{δ}$ generated

Marta Frankowska, Andrzej Nowik (2011)

Colloquium Mathematicae

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We prove that the ideal (a) defined by the density topology is not $G_{δ}$ generated. This answers a question of Z. Grande and E. Strońska.

On AP spaces in concern with compact-like sets and submaximality

Mi Ae Moon, Myung Hyun Cho, Junhui Kim (2011)

Commentationes Mathematicae Universitatis Carolinae

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The definitions of AP and WAP were originated in categorical topology by A. Pultr and A. Tozzi, Equationally closed subframes and representation of quotient spaces, Cahiers Topologie Géom. Différentielle Catég. 34 (1993), no. 3, 167-183. In general, we have the implications: $T_{2} \Rightarrow K C \Rightarrow U S \Rightarrow T_{1}$ , where $K C$ is defined as the property that every compact subset is closed and $U S$ is defined as the property that every convergent sequence has at most one limit. And a space is called submaximal if every dense subset...

The order topology for a von Neumann algebra

Emmanuel Chetcuti, Jan Hamhalter, Hans Weber (2015)

Studia Mathematica

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The order topology $τ_{o} (P)$ (resp. the sequential order topology $τ_{o s} (P)$ ) on a poset P is the topology that has as its closed sets those that contain the order limits of all their order convergent nets (resp. sequences). For a von Neumann algebra M we consider the following three posets: the self-adjoint part $M_{s a}$ , the self-adjoint part of the unit ball $M ¹_{s a}$ , and the projection lattice P(M). We study the order topology (and the corresponding sequential variant) on these posets, compare the order topology...

The quasi Isbell topology on function spaces

D. N. Georgiou, A. C. Megaritis (2015)

Colloquium Mathematicae

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In this paper, on the family (Y) of all open subsets of a space Y we define the so called quasi Scott topology, denoted by $τ_{q S c}$ . This topology defines in a standard way, on the set C(Y,Z) of all continuous maps of the space Y to a space Z, a topology $t_{q I s}$ called the quasi Isbell topology. The latter topology is always larger than or equal to the Isbell topology, and smaller than or equal to the strong Isbell topology. Results and problems concerning the topology $t_{q I s}$ are given.

Function space topologies deriving from hypertopologies and networks

A. Di Concilio, A. Miranda (2001)

Bollettino dell'Unione Matematica Italiana

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In un progetto di generalizzazione delle classiche topologie di tipo «set-open» di Arens-Dugundji introduciamo un metodo generale per produrre topologie in spazi di funzioni mediante l'uso di ipertopologie. Siano $X$ , $Y$ spazi topologici e $C (X, Y)$ l'insieme delle funzioni continue da $X$ verso $Y$ . Fissato un «network» $α$ nel dominio $X$ ed una topologia $τ$ nell'iperspazio $C L (Y)$ del codominio $Y$ si genera una topologia $τ_{α}$ in $C (X, Y)$ richiedendo che una rete $\{f_{λ}\}$ di $C (X, Y)$ converge in $τ_{α}$ ad $f \in C (X, Y)$ se e solo se la rete $\{\bar{f_{λ} (A)}\}$ converge...

On spaces with the ideal convergence property

Jakub Jasinski, Ireneusz Recław (2008)

Colloquium Mathematicae

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Let I ⊆ P(ω) be an ideal. We continue our investigation of the class of spaces with the I-ideal convergence property, denoted (I). We show that if I is an analytic, non-countably generated P-ideal then (I) ⊆ s₀. If in addition I is non-pathological and not isomorphic to $I_{b}$ , then (I) spaces have measure zero. We also present a characterization of the (I) spaces using clopen covers.

Locally solid topologies on spaces of vector-valued continuous functions

Marian Nowak, Aleksandra Rzepka (2002)

Commentationes Mathematicae Universitatis Carolinae

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Let $X$ be a completely regular Hausdorff space and $E$ a real normed space. We examine the general properties of locally solid topologies on the space $C_{b} (X, E)$ of all $E$ -valued continuous and bounded functions from $X$ into $E$ . The mutual relationship between locally solid topologies on $C_{b} (X, E)$ and $C_{b} (X)$ $(= C_{b} (X, ℝ))$ is considered. In particular, the mutual relationship between strict topologies on $C_{b} (X)$ and $C_{b} (X, E)$ is established. It is shown that the strict topology $β_{σ} (X, E)$ (respectively $β_{τ} (X, E)$ ) is the finest $σ$ -Dini topology...