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Page 38
... F } is an open covering of F. By the Heine - Borel Theorem there is a finite subcollection { O ,, ... , On } which covers F. Let d 8 ) . Then 8 is positive . Given two points y and z in F such that ly - z < 8 , the point y must belong ...
... F } is an open covering of F. By the Heine - Borel Theorem there is a finite subcollection { O ,, ... , On } which covers F. Let d 8 ) . Then 8 is positive . Given two points y and z in F such that ly - z < 8 , the point y must belong ...
Page 131
H. L. Royden. function f defined on X such that 0 ≤f ≤1 on X while f = 0 on A and f = 1 on B. If X is any set of points and F is any collection of real - valued ... Let f be corresponding to each Sec . 3 ] THE SEPARATION AXIOMS 131.
H. L. Royden. function f defined on X such that 0 ≤f ≤1 on X while f = 0 on A and f = 1 on B. If X is any set of points and F is any collection of real - valued ... Let f be corresponding to each Sec . 3 ] THE SEPARATION AXIOMS 131.
Page 132
... Let { U , } be the family constructed in ( b ) with U1 the real - valued function on X defined by f ( x ) = inf { r : x ε U , } . Then ƒ is a continuous function , 0 ≤ƒ≤ 1 , with ƒ = 0 on F = 1 on 0 . and f d . Let X be a Hausdorff ...
... Let { U , } be the family constructed in ( b ) with U1 the real - valued function on X defined by f ( x ) = inf { r : x ε U , } . Then ƒ is a continuous function , 0 ≤ƒ≤ 1 , with ƒ = 0 on F = 1 on 0 . and f d . Let X be a Hausdorff ...
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A₁ absolutely continuous axiom B₁ Baire set Banach space Borel equivalent Borel sets Borel subset bounded linear functional called Cauchy sequence closed sets cluster point Co(X compact Hausdorff space compact space continuous function continuous real-valued functions convex set Corollary countable collection Daniell integral definition denote E₁ E₂ elements finite measure finite number following proposition function defined function f ƒ and g given Hausdorff space Hence homeomorphism infinite L₁ Lebesgue measure Lemma Let f Let ƒ linear manifold measurable function measurable sets measure algebra measure space measure zero metric space monotone natural numbers nonempty nonnegative measurable function o-algebra o-finite one-to-one open intervals open set outer measure point of closure Problem Proof Prove rational numbers semicontinuous set function set of finite set of measure Show simple function topological space unique vector lattice x₁