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Description: this is solved question, please type the answer i can't read handwriting In ametric space, a subset i's Gompact id y everysequence of di'stinct elements imsubsequence with limit inIn discrete mehic spare,it hasa convengentthe subset.I8 Let set Nn s i

In a metric space, a subset is compact iff every subsequnce of distinct elements in it has a…

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Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter1: Fundamentals

Section1.7: Relations

Problem 13E: 13. Consider the set of all nonempty subsets of . Determine whether the given relation on is...

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13. Consider the set of all nonempty subsets of . Determine whether the given relation on is reflexive, symmetric or transitive. Justify your answers. a. if and only if is subset of . b. if and only if is a proper subset of . c. if and only if and have the same number of elements.
Let S be a nonempty subset of R that is bounded below. Prove that inf S= -sup{-s:s∊S}
3. a) We know that the set S = {1/n : n ∈ N} is not compact because 0 is a limit point of S that is not in S. To see the non-compactness of S in another way, find an open cover of S that does not have a finite subcover! b) Is the Cantor set closed? Is it compact? Explain!
We know that the set S = {1/n : n ∈ N} is not compact because 0 is a limit point of S that is not in S. To see the non-compactness of S in another way, find an open cover of S that does not have a finite subcover!
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6) What is the limit of T(n) as n -> ∞?
Assuming that R has the euclidean topology. Let D be set of dense subsets ofR. Show that there is subset of D which is equivalent with P, the set of all irrationalnumbers
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Let S be an nonempty open connect subset of R^2. What is the number of possible continuous onto functions from \overline{S} to the set of all rational numbers Q ?
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Let S be an ordered set. Let B ⊂ S be bounded (above and below). Let A ⊂ B be a nonempty subset. Suppose all of the inf's and sup's exist. Show that: inf B ≤ inf A ≤ sup A ≤ sup B.