a) d(x,x)=0

b) d(x,y)=d(y,x)

c) there exists x,y so that d(x,y)=1

d) d(x,z) = d(x,y) + d(y,z)

e) more than one answer is not a part of the definition of a metric

f(x):R→R is continuous at x0 if

a) ∃ε>0,s.t.∀δ>0and∀x,|x−x0|<δ⇒|f(x)−f(x0)|<ε

b) ∀ε>0,∃δ>0s.t. ∀x,|x−x0|<δ⇒|f(x)−f(x0)|<ε

c) ∀ε > 0,∃δ > 0 s.t. ∀x,d′(x,x0) < δ ⇒ d′(f(x),f(x0)) < ε

d) All of the above

e) More than one answer from a, b and c holds, but not all three

Which of the following hold?

a) f(C1 ∩ C2) = f(C1) ∩ f(C2)

b) f(C1 ∪ C2) = f(C1) ∪ f(C2)

c) f(C1 ∩ C2) ⊆ f(C1) ∩ f(C2)

d) Only a) and b)

e) Only b) and c) f) Only a) and c)

Which of the following are open sets?

a) An open interval in R

b) An open diamond in R2

c) An open diamond in R2taxicab

d) Only a) and b)

e) Only a) and c)

f) All of the above: a), b) and c)

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)?

a) Take δ = ε

b) Takeδ=ε−d(x,y)

c) Takeδ=d(x,y)−ε

d) Any of the above are possible

e) It is not possible to choose a δ successfully because the set is not open

What sets are open in the discrete metric on X?

a) No sets are open

b) Only circles of radius 1 are open

c) Any subset of X is open

d) Other

Which of the following are correct interpretations of open in this topology course?

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.

b) (a,b) is always open

c) A set is open if it is in the topology Tau

d) a) and b)

e) a) and c)

Which of the following collections of sets are in every topology?

1. finite unions

2. finite intersections

3. arbitrary unions

4. arbitrary intersections

a) 1. and 4.

b) 2. and 3.

c) 1. 2. and 3.

d) 1. 2. and 4.

e) 2. 3. and 4.

Given a collection of sets, how do we generate the smallest topology containing the sets?

a) Take finite unions and arbitrary intersections and add those to the collection

b) If necessary, add X to the collection

c) Both of the above

d) Other

In linear algebra, a basis is:

a) An efficient way to represent a vector space

b) A maximum linearly independent set

c) A minimum spanning set

d) All of the above

e) Other