Let A be a subset of a topological set X. Then the closure of A is the union of of A and its set of accumulation points, ie. Ā = A U A'.
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What is the interior of a set, denoted A°?
The union of all open subsets of A.
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When is subset compact?
If every open cover of the subset is reducible to a finite cover.
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Give an example of a compact interval.
Every closed and bounded interval [a,b] on the real line R.
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What is an open set?
A set is open if and only if each of its points is an interior point.
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What is an interior point?
A point p e A is an interior point of A if and only if p belongs to some open interval Sp which is contained in A : p e Sp ¢ A.
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When is a set a closed set?
It is a closed set if and only if its complement, A‹ , is an open set; or if the set contains each of its accumulation points.
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What is an accumulation (or limit) point?
A point p e R is an accumulation point of A if and only if every open set G containing p contains a point of A different from p; ie. G open, p e G implies A ∏ (G \ {p}) ≠ Ø.
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What is the set of accumulation points?
The derived set, A'.
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When is a sequence <an : n e N> a Cauchy sequence?
If and only if for every € > 0, there exists a positive integer n• such that n,m > n• implies )an - am( < €.
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In words, what is the definition of a Cauchy sequence?
If terms of the sequence become arbitrarily close to each other as n gets larger.
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What is the nested interval property?
Let I¡ = [a¡, b¡], I2 = [a2, b2],… be a sequence of nested, closed (bounded) intervals. Then there exists at least one point common to every interval, π I¡ ≠ ø.
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What is the least upper bound axiom?
If A is a set of real numbers bounded from above, then A has a least upper bound, ie. sup (A) exists.
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What is the Bolzano-Weirestrass theorem?
Let A be a bounded, infinite set of real numbers. Then A has at least one accumulation point.
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What is a subset?
A set A is a subset of a set B, or B is a superset of A, written as A ¢ B, if and only if each element in A belongs to B; ie. x e A implies x e B.
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What is the Heine-Borel theorem?
Let A = [c,d] be a closed and bounded interval, and let G = {G¡ : i e I} be a class of open intervals which covers A. Then G contains a finite subclass, say {G¡1, …, G¡m} which also covers A.
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Other cards in this set
Card 2
Front
What is the closure of a set, denoted Ā?
Back
Let A be a subset of a topological set X. Then the closure of A is the union of of A and its set of accumulation points, ie. Ā = A U A'.
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