# Is it true that, given a set E, every point in the set of boundary points of E is an accumulation point of the set of boundary points?

Suppose $$E\subset \mathbb{R}^n.$$ Is it true that for all $$x\in\partial E,$$ $$x$$ is an accumulation point of $$\partial E$$?

The reason I think this is true (despite my feeling it is false) is that we have $$\partial E=\{ p \in \mathbb{R}^n, \forall r>0 \text{ one has } B(p,r)\cap E\neq\emptyset \text{ and } B(p,r)\cap E^C \neq \emptyset \}$$

which implies, with some extra reasoning, that every ball about every point in $$\partial E$$ Intersects $$S-\{p\}.$$

• What is the boundary of the set $\{0\}$ in $\mathbb R$? What are the accumulation points? – GEdgar Feb 25 at 22:15
• @GEdgar the boundary is 0 and there are no accumulation points. Got it. Counter example. Thanks. – Jake Feb 25 at 22:17

$$E=(0,1) \subseteq \mathbb R$$ only has two boundary points and these are not accumulation points of that finite boundary....