September 2, 2017

Download A contrastive phonological analysis of English and Persian: by Luṭf Allāh Yār Muḥammadī PDF

By Luṭf Allāh Yār Muḥammadī

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9 Theorem (Total extension of preorders) extension to a total preorder. Any preorder has a compatible 16 Chapter 1. Odds and ends Proof : Let R be a preorder (reflexive transitive binary relation) on the set X. Let E be the set of preorders that compatibly extend R, and let E be partially ordered by inclusion (as subsets of X × X). Note that E contains R and so is nonempty. Let C be a nonempty chain in E. We claim the relation U = {S : S ∈ C} is an upper bound for C in E. Clearly U is reflexive and extends R.

The sets that are countable intersections of open sets or countable unions of closed sets are important enough that they have been given two special, albeit curious, names. 26 Chapter 2. 3 Definition • a Gδ -set, or simply a Gδ , if it is a countable intersection of open sets. • an Fσ -set, or simply an Fσ , if it is a countable union of closed sets. 2 1 The example (0, 1] = ∞ n=1 n , 1 = simultaneously a Gδ - and an Fσ -set. 2 ∞ n=1 0, 1 + 1 n shows that a set can be Neighborhoods and closures Let (X, τ) be a topological space, and let A be any subset of X.

4. If x < ω1 , then y ∈ Ω : x ≤ y ≤ ω1 is uncountable. 5. Every nonempty subset of Ω has a least upper bound. 6. A nonempty subset of Ω\{ω1 } has a least upper bound in Ω\{ω1 } if and only if it is countable. In particular, the least upper bound of every uncountable subset of Ω is ω1 . Proof : Let (X, ≤) be an uncountable well ordered set, and consider the set A of elements x of X such that the initial segment I(x) = {y ∈ X : y ≤ x} is uncountable. Without loss of generality we may assume A is nonempty, for if A is empty, append a point y to X, and extend the ordering ≤ by x ≤ y for all x ∈ X.

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