Local Connectedness Made Uniform

Baboolal, Dharmanand

doi:10.1007/978-94-017-2529-3_23

Cited by 8 publications

(12 citation statements)

References 9 publications

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“…The counterpart notion of property S in Frm appears in [1] and the proof of the following result is identical to that of Proposition 2.4 therein with minor necessary changes taking into account countable joins. Lemma 1.…”

Section: !%%mentioning

confidence: 83%

“…Using the uniformly locally connected reection of a locally connected uniform frame established in [1] we construct the locally connected reection of a locally connected uniform σ-frame. Recall from [1] that for a locally connected uniform frame (N, ξ) its uniformly locally connected reection is given by (N, ξ) where ξ ξ with ξ = {A ∈ cov L : B A for some B ∈ ξ} where B = {x ∈ L : x is a component of some b ∈ B}. Here a component of b is a maximal connected element x b.…”

Section: !%%mentioning

confidence: 99%

“…The construction of the uniformly locally connected coreection of an arbitrary uniform space (appearing in the paper by Gleason [10]) has been realised in the pointfree setting by Baboolal in [1] for locally connected frames that possess a uniformity. With the introduction of structured σ-frames in [11] and [14] together with the σ-frame analogue of local connectedness in [12], it is the main purpose of this paper to establish Gleason's construction in the uniform σ-frame setting where the underlying σ-frame of a uniform σ-frame is locally connected.…”

Section: Introductionmentioning

confidence: 99%

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Uniformly locally connected uniform σ-frames

Naidoo

2008

Acta Math Hung

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Section: !%%mentioning

confidence: 83%

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confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Uniformly locally connected uniform σ-frames

Naidoo

2008

Acta Math Hung

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“…Uniform local connectedness with respect to along. We introduce the notion of uniform local connectedness with respect to along for uniform frames and prove that for dense surjections, this concept coincides with uniform local connectedness introduced by Baboolal in [1]. The significance of this concept is given in Section 3 where our approach to complete uniform spreads has a bearing on uniform local connectedness with respect to along.…”

Section: Introductionmentioning

confidence: 83%

“…The second statement is a consequence of a result of Baboolal [1]. Assume that h : L → M is a surjective homomorphism and that M is uniformly locally connected with respect to L along h. Then given A ∈ UM, we have h[B] ≤ A for some B ∈ UL.…”

Section: Introductionmentioning

confidence: 96%

Aspects of complete uniform spreads in frames

Siweya¹

2002

International Journal of Mathematics and Mathematical Sciences

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We give a new notion of a complete uniform spread in terms of a relative kind of uniform local connectedness. Properties of this type of (uniform) connectedness are discussed. It is also shown that our concept of complete spread is equivalent to that of Hunt (1982).2000 Mathematics Subject Classification: 54D15, 54E15.1. Introduction. The concept of a complete uniform spread together with its completion was introduced into the category Unif of uniform spaces and uniformly continuous functions by Hunt [6] (see also [7]). In [11], we introduced these concepts into the category UniFrm of uniform frames and uniform homomorphisms.Section 2 is devoted to properties of uniform local connectedness with respect to along. We also show how the Banaschewski-Pultr uniform frame completion CL of a uniform frame L can be used to obtain point-free analogies of some well-known topological results. Section 3 is devoted to complete spreads. Hunt [8] defines a uniform spread f : (X, ᐁ) → (Y , ) to be complete if (X, ᐁ) is a complete uniform space where ᐁ is a spread uniformity generated by f . We give a new definition of a complete uniform spread in terms of the concept of uniform local connectedness with respect to along. It is then shown that such a definition is equivalent to that of Hunt (1982).We assume familiarity with general knowledge of frames, especially [9] and uniform frames according to [3,4,5]. Briefly, a cover of a frame L is a subset A ⊆ L such that A = e, the top element of L. A cover U of L is said to refine a cover V of L if for each x ∈ U there is y ∈ V such that x ≤ y. We write U ≤ V . Given a cover A of L and x ∈ L, we define the A-star of x to be the element

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A spread extension associated with Madden-type generators

Siweya

2006

Acta Math Hung

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A frame homomorphism f : L → M between locally connected frames is called a localic spread if u∈L S u is a basis for M , whereis a component of h(u)". Maddentype generators and relations are applied on L to form a freely generated frame CM induced by j : M → CM leading to a spread extension j • f : L → CM of f .In this article, we discuss properties of a local spread extension (which is not complete) between locally connected frames.

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Local Connectedness Made Uniform

Cited by 8 publications

References 9 publications

Uniformly locally connected uniform σ-frames

Uniformly locally connected uniform σ-frames

Aspects of complete uniform spreads in frames

A spread extension associated with Madden-type generators

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