n3 - Atsushi Inoue ECG 765: Mathematical Methods for...

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ECG 765: Mathematical Methods for Economics Fall 2009 Lecture Notes 3 Open and Closed Sets In optimization problems we often assume that constraint sets are compact for a reason that we will explain later. In the Euclidean space a set is compact if and only if it is closed and bounded. We will study these concepts. Outline A. Open Sets B. Closed Sets A. Open Sets Deﬁnition. The ball B r ( x 0 ) with center x 0 ∈ < n and radius r > 0 is deﬁned to be the set of all x ∈ < n such that k x - x 0 k < ε : B r ( x 0 ) = { x ∈ < n : k x - x 0 k < ε } . Deﬁnition. A set S is open in < m if for each x S , there exists an open ε -ball around x completely contained in S : x S there is an ε > 0 such that B ε ( x ) S. Deﬁnition. (a) A B , spoken “A union B”, is the set of all elements that are either in A or in B (or in both): A B = { x : x A or x B } . (b)

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This note was uploaded on 10/22/2009 for the course ECG 765 taught by Professor Fackler during the Fall '07 term at N.C. State.

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n3 - Atsushi Inoue ECG 765: Mathematical Methods for...

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