# I need to write a set of infinities elements and only one point of accumulation.

And also I need to write a set unbounded that has only one point of accumulation and all others isolated point. I know that the set of numbers naturals has infinite points but don't have any point of accumulation, but I don't know which is those sets of the question. I have like the definition of point accumulation :

For the set X ⊆ R, the neighborhood of a Vε (a) ∩X-{a} ≠ ∅, ε > 0 is the variation of neighborhood

• You want a set of real numbers with exactly one accumulation point? Can you think of a decreasing set of real numbers that converges to $0$, say? – lulu Mar 30 '18 at 14:17
• Do you have the definition of "accumulation point"? – Eric Towers Mar 30 '18 at 14:25
• So, I can write something like this: 0 < = x < 1 ?? – Juliana Rodrigues Mar 30 '18 at 14:26
• For the set X ⊆ R, the neighborhood of a Vε (a) ∩X-{a} ≠ ∅ **ε > 0, is the variation of neighborhood ** – Juliana Rodrigues Mar 30 '18 at 14:38
• The English term is "isolated point", not "insulated point". Because somebody many years ago said so. – DanielWainfleet Mar 30 '18 at 17:33

Any convergent sequence form a set with only a point of accumulation so you can take for exemple $A = \{ \frac{1}{n} ; n \in \mathbb{N} \}$
and an unbounded set with the same property is $A \cup \mathbb{N}$