Is there any open set on zero dimensional Euclidean space $\mathbb{R}^0$?

Question

I was asking this as I'm going through the definition of small inductive dimension. In my opinion, since $\mathbb{R}^0=\{0\}$(I suppose?), so one cannot find a neighborhood for $0\in \mathbb{R}^0$, so there's no possible open set on it... But I'm not sure. Anybody could help me? Thanks!

Answer 1

That $\mathbb{R}^0 = \{0\}$ I agree with (makes sense from a linear space point of view e.g.) or also in set-theory.

This space has two open sets $\{0\}$ and $\emptyset$ and both are clopen. So $\operatorname{ind}(\mathbb{R}^0) = 0$.

Answer 2

$\mathbb{R}^0$ and the empty set are open (as always). Since there is only one element in the space, there is no other subset, so the only open sets are the aforementioned ones (the trivial sets).

Answer 3

When your topological space is $X=\{0\}$ The only open sets are $\phi$ and $X$.

Thus if you are looking for an open neighborhood of $\{0\}$ , you get $X =\{0\}$