What are the interior and limit points of the given subset of n-dimensional Euclidean space?

Question

Considering metric topology and giving the set E subspace topology in the Euclidean Space. Given E={(a$_{1}$, a$_{2}$,...,a$_{n-1}$, 0) | a$_{i}\in\mathbb{R}$} $\subset$$\mathbb{R}^{n}$. I want to determine the limit and interior points of the set E.

Considering the case n=2, E is just the horizontal component $\mathbb{R}$ and since every point is a limit point as well as interior point for $\mathbb{R}$ I feel like the whole set is both interior and the limit points is E $\cup$ {$\infty$}

Any comments would be appreciated.

Answer 1

You're almost there, but you need to be careful. Look back at your example when $n =2$ and think about the definition of interior point. A point $(x,0) \in E$ is an interior point if there is a small ball $B \subset \mathbb R^2,$ centered at $(x,0)$ with the property that $B \subset E$. But this is impossible. Also note that $\infty$ is not an element of $\mathbb R,$ so that you shouldn't regard this symbol as a limit point. A similar argument holds for the general case.