1. ## Looking the name

If $x\in A$ verifies that $\forall\epsilon:B(x,\epsilon)\cap A\ne\varnothing,$ how is it called?

Thank you!

2. Originally Posted by Connected
If $x\in A$ verifies that $\forall\epsilon:B(x,\epsilon)\cap A\ne\varnothing,$ how is it called?
Is there a typo here?
Because this is always true since $x\in B(x,\epsilon)\cap A$.

3. No, no typo, it's just that I want to know how the point $x$ is called.

4. Originally Posted by Connected
No, no typo, it's just that I want to know how the point $x$ is called.
Then I want to learn this too.

5. Haha, okay, perhaps another known example:

$x\in A$ is said to be an interior point of $A$ if $\exists\delta>0:B(x,\delta)\subseteq A.$

6. $x\in X$ is said to be an accumulation (limit) point of $A$
provided that $(B(x,\varepsilon)\backslash\{x\})\cap A\neq\emptyset$ for every $\varepsilon>0$.

May be you are looking for this one?

7. Got it, the concept is "adherent point."

8. Originally Posted by Connected
Got it, the concept is "adherent point."
What you mean to say is that $x\in A$ or etc.

9. I think the other name that has is "closure point," does that make sense?