On strict and simple type extensions

Mohan Tikoo

International Journal of Mathematics and Mathematical Sciences, 1998, vol. 21, 1-9

Abstract:

Let ( Y , τ ) be an extension of a space ( X , τ ′ ) ⋅ p ∈ Y , let 𝒪 y p = { W ∩ X : W ∈ τ , p ∈ W } . For U ∈ τ ′ , let o ( U ) = { P ∈ Y : U ∈ 𝒪 y p } . In 1964, Banaschweski introduced the strict extension Y # , and the simple extension Y + of X (induced by ( Y , τ ) ) having base { o ( U ) : U ∈ τ ′ } and { U ∪ { p } : p ∈ Y , and U ∈ O y p } , respectively. The extensions Y # and Y + have been extensively used since then. In this paper, the open filters ℒ p = { W ∈ τ ′ : W ⫆ int x cl x ( U ) for some U ∈ 𝒪 y p } , and 𝒰 p = { W ∈ τ ′ : int x cl x ( W ) ∈ 𝒪 y p } = { W ∈ τ ′ : int x cl x ( W ) ∈ ℒ p } = ∩ { 𝒰 : 𝒰 is an open ultrafilter on X , 𝒪 y p ⊂ 𝒰 } on X are used to define some new topologies on Y . Some of these topologies produce nice extensions of ( X , τ ′ ) . We study some interrelationships of these extensions with Y # , and Y + respectively.

Date: 1998
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Persistent link: https://EconPapers.repec.org/RePEc:hin:jijmms:450364

DOI: 10.1155/S0161171298000349

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