Which of the following are not part of the definition of a metric? 
<br>a) d(x,x)=0 
<br>b) d(x,y)=d(y,x) 
<br>c) there exists x,y so that d(x,y)=1 
<br>d) d(x,z) = d(x,y) + d(y,z) 
<br>e) more than one answer is not a part of the definition of a metric 
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f(x):R→R is continuous at x0 if
<br>a) ∃ε>0,s.t.∀δ>0and∀x,|x−x0|<δ⇒|f(x)−f(x0)|<ε 
<br>b) ∀ε>0,∃δ>0s.t. ∀x,|x−x0|<δ⇒|f(x)−f(x0)|<ε
<br>c) ∀ε > 0,∃δ > 0 s.t. ∀x,d′(x,x0) < δ ⇒ d′(f(x),f(x0)) < ε 
<br>d) All of the above
<br>e) More than one answer from a, b and c holds, but not all three
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<br>Which of the following hold?
<br>a) f(C1 ∩ C2) = f(C1) ∩ f(C2) 
<br>b) f(C1 ∪ C2) = f(C1) ∪ f(C2) 
<br>c) f(C1 ∩ C2) ⊆ f(C1) ∩ f(C2)
<br>d) Only a) and b) 
<br>e) Only b) and c) f) Only a) and c)
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Which of the following are open sets?
<br>a) An open interval in R 
<br>b) An open diamond in R2
<br>c) An open diamond in R2taxicab 
<br>d) Only a) and b)
<br>e) Only a) and c) 
<br>f) All of the above: a), b) and c)
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In a successful proof that Bd(x,ε) is open, if y ∈ Bd(x,ε), what δ can you take for Bd(y, δ) to be in Bd(x, ε)?
<br>a) Take δ = ε 
<br>b) Takeδ=ε−d(x,y)
<br>c) Takeδ=d(x,y)−ε 
<br>d) Any of the above are possible
<br>e) It is not possible to choose a δ successfully because the set is not open 
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What sets are open in the discrete metric on X?
<br>a) No sets are open 
<br>b) Only circles of radius 1 are open
<br>c) Any subset of X is open 
<br>d) Other

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Which of the following are correct interpretations of open in this topology course? 
<br>a) A set U in a metric space is open if for every x in U there exists epsilon>0 so that B(x,epsilon) is contained in U. 
<br>b) (a,b) is always open 
<br>c) A set is open if it is in the topology Tau 
<br>d) a) and b) 
<br>e) a) and c) 
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Which of the following collections of sets are in every topology?
<br>1. finite unions 
<br>2. finite intersections 
<br>3. arbitrary unions 
<br>4. arbitrary intersections
<br>a) 1. and 4. 
<br>b) 2. and 3. 
<br>c) 1. 2. and 3. 
<br>d) 1. 2. and 4. 
<br>e) 2. 3. and 4. 
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Given a collection of sets, how do we generate the smallest topology containing the sets? 
<br>a) Take finite unions and arbitrary intersections and add those to the collection 
<br>b) If necessary, add X to the collection 
<br>c) Both of the above 
<br>d) Other 
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In linear algebra, a basis is: 
<br>a) An efficient way to represent a vector space 
<br>b) A maximum linearly independent set 
<br>c) A minimum spanning set 
<br>d) All of the above 
<br>e) Other