# What can you say about interior points of a non empty subset of real numbers?

Given that A is a non-empty subset of real numbers, if I(A) denotes the set of interior points of A; then I (A) is:-

a) empty. b) singleton. c) a finite set containing more than one element. d) countable but not finite.

I know that the largest open set contained in A is called interior of A. Also a point is said to be interior point of A if we have an open ball of finite radius contained in A.

My trouble is I am not able to figure out the interpretation from the given statement as it is not given anything else beside non empty subset.

• None of these options has to hold. Take $A=\mathbb{R}$. The interior is all of $\mathbb{R}$, which is nonempty, not a singleton, infinite, and uncountable.
– kccu
Jan 10, 2016 at 5:03
• But can I say option d) as answer if I dont take A to be R? But again problem occurs on open intervals ! For ex. (0,1) is uncountable. Jan 10, 2016 at 5:13
• No, take for example $A=[0,1]$. The interior $(0,1)$ is uncountable. In fact, if $A$ has nonempty interior, $I(A)$ contains an open interval and is thus uncountable.
– kccu
Jan 10, 2016 at 5:15
• Are you sure you don't mean "boundary points" and not "interior points"?
– user2469
Jan 10, 2016 at 5:31

The interior of any subset $S$ of $\mathbb R$ is open in $\mathbb R$, so it is either empty or a countable union of open balls in $\mathbb R$.