Is it possible for an element in a topology to be both boundary as well as limit point for a set?

e.g. S = {1,2,3}

Let us say 0 is null set then

T = {0, {1,2},{3},S} defines a toplogy on it.

Let is say A={1,3}

Can I have a case where a particular element of A is both limit point as well as boundary point of A.

TAKE THE CASE GIVEN ABOVE AS AN EXAMPLE ONLY.

## Can an element be both boundary as well as limit point in a

But before let's just in case go back and remember what actually boundary and limit point are.

*The*

A point s in S is called

**boundary**of A, bd(A) is the set of all s in S such that every open set containing s intersects both A and complement of A.A point s in S is called

**limit point of A**if every open set that contains s also contains at least one point of A other than s.So let's consider simple example with the set of real numbers with its standard topology. Let A be the interval (0,1). In this case we have

bd(A) = {0,1}

A' = [0,1].

Here A' -

**derived set**- the set of all the limit points of A.

Therefore two elements of A, numbers 0 and 1 are both boundary and limit points.

In your case where

I see thatS = {1,2,3}

T = {0, {1,2},{3},S}

A={1,3}

bd(A)=0

A'=0,

so it is not really felicitous example.