Understanding Limit Points and Isolated Points in Different Sets

[Unassuming]

Could somebody throw their hat in with me on these?

a. $$\{1- \frac{(-1)^n}{n}:n \in \mathbb{N} \}$$

Limit point = 1 , Isolated points are equal to the whole set

b. $$\{ (-1)^n + \frac{1}{n} : n \in \mathbb{N} \}$$

Limit points are -1 and 1, Isolated points are the whole set

c. $$(0,1) \cup 2$$

Limit points are [0,1], Isolated point is 2.

d. $$\mathbb{N}$$

Limit points none, Isolated points are $$\mathbb{N}$$

e. $$\mathbb{R} \backslash \mathbb{Q}$$

Limit points are $$\mathbb{R}$$, Isolated points are none.

f. $$\mathbb{Q} \cap (0,1)$$

Limits points are [0,1], Isolated points are none

 

[morphism]

If by limit/isolated points you mean limit/isolated points in R with its usual topology, then I agree with what you have.

 

[Unassuming]

Yes, thanks morphism.

I meant usual topology and R1. thanks