Week 3; 3.3-4

Question 1

A point x€X is called a limit point of a set A if

Answer 1

Every ball about x contains a point of A distinct from x

Question 2

Other terms for limit point

Answer 2

Accumulation point
Cluster point

Question 3

Set of limit points of A is denoted

Answer 3

A’

Question 4

A point x€A’ if?
(From sequences)

Answer 4

And only if there is a sequence x_n of elements of A distinct from x which converges to x

Question 5

Proof of definition of limit point from sequences

Answer 5

Question 6

Relate a closed set on (X, d) to limit points

Answer 6

A set is closed on (X, d) if it contains all of its limit points

Question 7

Define closed set from sequences

Answer 7

A set A is closed in (X, d) <=> for all sequences (x_n) in A that converge in X we have

Question 8

Prove definition of closed set from sequences

Answer 8

Question 9

Prove that a closed ball in a metric space (X, d) is closed

Answer 9

Question 10

Prove that complement of closed set is open (+vice versa)

Answer 10

Question 11

3 properties of metric space regarding closed sets

Answer 11

Question 12

Example of set that is neither open nor closed

Answer 12

Question 13

The closure of a set A is

Answer 13

The intersection of all closed sets containing A

Question 14

Denote closure of set A

Answer 14

Question 15

Cl(A) is ?

Answer 15

The closure of set A
The smallest (wrt set inclusion) closed set containing A

Question 16

Relate A cl(A) and A’

Answer 16

Question 17

Prove

Answer 17

Question 18

Define Cl(A) from sequences

Answer 18

Question 19

Prove

Answer 19

Question 20

Define Cl(A) from nbhds

Answer 20

Question 21

Prove

Answer 21

Question 22

Prove

Answer 22