METRIC SPACES 2 EXAM QUESTIONS
AND ANSWERS
What is the definition of an open set? - ANSWER Let (X,d) be a metric space.
The set U subset X is open in X concerning 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? - ANSWER The empty set. The total
space X.
What can we say about open balls and open sets? (2) - ANSWER Every open
ball Br(x) is open in X.
A subset U is open in (X,d) iff it is a union of open balls.
Give four examples of open sets - 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? - ANSWER open
intervals!
Give three sets on the Euclidean line that are not open. - 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? - ANSWER the set X
and the metric d
Give the theorem linking open sets and Lipschitz equivalency. - 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? - ANSWER Have
the dame open sets
Give three main properties of open sets - 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 -
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? - 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? - ANSWER The subset U^o of all inferior point.
In terms of the interior points, what makes the set U open? - ANSWER The set
U is open in X iff U^o = U
Give two examples of sets that have no interior - 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
Give four properties of interiors - ANSWER Given any two subsets of X, U,V,
the following hold
1. U subset V implies that Uo subset Vo
2. (Uo)o = Uo
3. Uo is open in X
4. Uo is the largest subset of U that is open in X.
What is another was to characterise the interior? - ANSWER The union of all
the open sets of X.
What is the initial definition of a closed set? - ANSWER The subset of X V is
closed in X iff X\V is open in X
AND ANSWERS
What is the definition of an open set? - ANSWER Let (X,d) be a metric space.
The set U subset X is open in X concerning 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? - ANSWER The empty set. The total
space X.
What can we say about open balls and open sets? (2) - ANSWER Every open
ball Br(x) is open in X.
A subset U is open in (X,d) iff it is a union of open balls.
Give four examples of open sets - 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? - ANSWER open
intervals!
Give three sets on the Euclidean line that are not open. - 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? - ANSWER the set X
and the metric d
Give the theorem linking open sets and Lipschitz equivalency. - 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? - ANSWER Have
the dame open sets
Give three main properties of open sets - 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 -
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? - 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? - ANSWER The subset U^o of all inferior point.
In terms of the interior points, what makes the set U open? - ANSWER The set
U is open in X iff U^o = U
Give two examples of sets that have no interior - 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
Give four properties of interiors - ANSWER Given any two subsets of X, U,V,
the following hold
1. U subset V implies that Uo subset Vo
2. (Uo)o = Uo
3. Uo is open in X
4. Uo is the largest subset of U that is open in X.
What is another was to characterise the interior? - ANSWER The union of all
the open sets of X.
What is the initial definition of a closed set? - ANSWER The subset of X V is
closed in X iff X\V is open in X
