Metric Spaces 2 UPDATED ACTUAL
Exam Questions and CORRECT Answers
What is the definition of an open set? - CORRECT ANSWER - Let (X,d) be a metric
space.
The set U subset X is open in X with respect to d iff for all u in U there exists r=r(u)>0 such that
Br(a) subset X.
What are two immediate open sets? - CORRECT ANSWER - The empty set. The total
space X.
What can we say about open balls and open sets? (2) - CORRECT ANSWER - Every open
ball Br(x) is open in X.
A subset U is open in (X,d) iff it isa union of open balls.
Give four examples of open sets - CORRECT ANSWER - 1. Any union of two open sets is
open.
2. The complement of a closed ball is open.
3. if (X,d) is discrete, any subset UsubsetX is open because B1/2(u) = {u} is a subset of U. (i.e. r
= 1/2)
4. Union of open balls
How can we interpret open sets on the Euclidean line? - CORRECT ANSWER - open
intervals!
Give three sets on the Euclidean line that are not open. - CORRECT ANSWER - 1. The
rationals are not open in R
2. The closed ball B-r(0) equipped with d2
3. The subset {(x,0) : x element R} equipped with d2
, What two factors are crucial in determining openness? - CORRECT ANSWER - the set X
and the metric d
Give the theorem linking open sets and Lipschitz equivalency. - CORRECT ANSWER -
Let (X, d1) and (X, d2) be two metric spaces on the same set X. If d1 and d2 are Lipschitz
equivalent then U subset of X is open wrt d1 IFF open wrt d2.
What do (Rn, d1) (Rn, d2) (Rn, dinfinity) have in common? - CORRECT ANSWER -
Have the dame open sets
Give three main properties of open sets - CORRECT ANSWER - 1. X and the empty set
are open in X
2. The union of open sets in open in X
3. The intersection of finite open sets is open in X
Give an example of an open interval whose intersection is NOT open in X - CORRECT
ANSWER - Vj, the open interval (-1/j, 1/j) in the Euclidean line R for every j = 1,2,...
Then the intersection is {0}, which is not open in R.
What is an interior point? - CORRECT ANSWER - The point u is an interior point iff
there exists epsilon>0 such that Bepsilon(u) is a subset of U.
What is the interior of U? - CORRECT ANSWER - The subset U^o of all inferior point.
In terms of the interior points, what makes the set U open? - CORRECT ANSWER - The
set U is open in X iff U^o = U
Give two examples of sets that have no interior - CORRECT ANSWER - 1. The set of
rationals Q, has no interior and so Q^o = empty
2. Rx = {(x,0)} of the Euclidean plan, (R2, d2) has no interior and so Rox is empty
Exam Questions and CORRECT Answers
What is the definition of an open set? - CORRECT ANSWER - Let (X,d) be a metric
space.
The set U subset X is open in X with respect to d iff for all u in U there exists r=r(u)>0 such that
Br(a) subset X.
What are two immediate open sets? - CORRECT ANSWER - The empty set. The total
space X.
What can we say about open balls and open sets? (2) - CORRECT ANSWER - Every open
ball Br(x) is open in X.
A subset U is open in (X,d) iff it isa union of open balls.
Give four examples of open sets - CORRECT ANSWER - 1. Any union of two open sets is
open.
2. The complement of a closed ball is open.
3. if (X,d) is discrete, any subset UsubsetX is open because B1/2(u) = {u} is a subset of U. (i.e. r
= 1/2)
4. Union of open balls
How can we interpret open sets on the Euclidean line? - CORRECT ANSWER - open
intervals!
Give three sets on the Euclidean line that are not open. - CORRECT ANSWER - 1. The
rationals are not open in R
2. The closed ball B-r(0) equipped with d2
3. The subset {(x,0) : x element R} equipped with d2
, What two factors are crucial in determining openness? - CORRECT ANSWER - the set X
and the metric d
Give the theorem linking open sets and Lipschitz equivalency. - CORRECT ANSWER -
Let (X, d1) and (X, d2) be two metric spaces on the same set X. If d1 and d2 are Lipschitz
equivalent then U subset of X is open wrt d1 IFF open wrt d2.
What do (Rn, d1) (Rn, d2) (Rn, dinfinity) have in common? - CORRECT ANSWER -
Have the dame open sets
Give three main properties of open sets - CORRECT ANSWER - 1. X and the empty set
are open in X
2. The union of open sets in open in X
3. The intersection of finite open sets is open in X
Give an example of an open interval whose intersection is NOT open in X - CORRECT
ANSWER - Vj, the open interval (-1/j, 1/j) in the Euclidean line R for every j = 1,2,...
Then the intersection is {0}, which is not open in R.
What is an interior point? - CORRECT ANSWER - The point u is an interior point iff
there exists epsilon>0 such that Bepsilon(u) is a subset of U.
What is the interior of U? - CORRECT ANSWER - The subset U^o of all inferior point.
In terms of the interior points, what makes the set U open? - CORRECT ANSWER - The
set U is open in X iff U^o = U
Give two examples of sets that have no interior - CORRECT ANSWER - 1. The set of
rationals Q, has no interior and so Q^o = empty
2. Rx = {(x,0)} of the Euclidean plan, (R2, d2) has no interior and so Rox is empty
