# Is a subset of a boundary closed?

I know that the boundary of a set $A \subset X$ is closed. Is it also true that a subset of a boundary is closed?

Thank you very much for your help

• No. Is every subset of a closed set closed? – user296602 Mar 13 '16 at 21:33
• @T.Bongers Not every closed set is a boundary. – egreg Mar 13 '16 at 21:35
• No, I know it is not true in the general case, just thought if there was an exception in the case of a boundary. – stensootla Mar 13 '16 at 21:35
• @egreg Indeed, but it's a good starting point to address whether it's reasonable to expect a yes answer. – user296602 Mar 13 '16 at 21:37
• Why would it be? subsets of closed sets aren't closed in general, so why would the set being a boundary j force its subsets to be closed. Consider A={(x,y)|y $le$ 2}. Its boundary is the line y=2. This line will have a subset (0,1)x {2}. Which is not closed. Consider any boundary A with more than one point and at least on limit point,a. Consider A/a. That can't be closed. – fleablood Mar 13 '16 at 22:28

No: $$\mathbb{Q} \subseteq \partial \mathbb{Q} = \mathbb{R}.$$
No. Consider the set $A=\{(x,y):x^2+y^2<1\}$ in two-space and the subset of the boundary $$T=\{(x,y):x^2+y^2=1,y>0\}$$ This set is not closed, because $(1,0)$ belongs to the closure, but not to $T$.
Let $A$ be a boundary with at least two points and at least one limit point. (Finding such a boundary is trivial. The unit circle in Euclidean plane will do.) Let $a$ be one of the limit points. Then $a$ is a limit point of the set A-{$a$}. So the set $A-{a}\subset A$ is not closed.