Question

Give an example of each of the following or explain why you think such a set could not exist.

(a) A nonempty set with no accumulation points and no isolated points
(b) A nonempty set with no interior points and no isolated points
(c) A nonempty set with no boundary points and no isolated points

Answer 1

Answer:

a.No

b.No

c.No

Step-by-step explanation:

a.No,Such set does not exist .A set of natural numbers is N

Every point of this set is an isolated point but no accumulation point

Accumulation point:It is defined as that point a of set Swhich every neighborhood contains infinitely many distinct point of set

$(a-\epsilon,e+\epsilon)\cap S-{a}\neq\phi$

Isolated point : it is defined as that point a of set S which neighborhood   does not contain any other point of set except itself

$(a-\epsilon0,a+\epsilon)\cap S-{a}=\phi$

Interior point of set :Let $a\in S$ .Then a is called interior point of set when its neighborhood is a subset of set S.

$a\in(a-\epsilon,e+\epsilon)\subset S$

When a set is uncountable then interior point exist it is  necessary for interior points existance .

Boundary points :Let $a\in S$ .If every non empty neighborhood of a  intersect S and complement of S.

Every member of  a set is a boundary point

b.No, such set does not exist .A non empty set with isolated point then the set have no interior points .By definition of interior point and isolated point .For example.set of natural numbers

c.No, Such set does not exist ,for example set of natural every point is an isolated point and boundary point.By definition of  boundary point and isolated point