Is $\zeta(-1)=0$ in the empty topological space?

Question

I have checked the zero ring properties I accrossed a nice property which states that The ring of continious real valued function is zero ring in empty topological space , Now in the zero ring we may assum that $1=0$ this mean that $\sum_{n=0}^{\infty} n=\sum_{n=0}^{\infty} 0=0$ yield to $\zeta(-1)=0$ such that Riemann zeta function is extend by analytic continuation to be defined in the whole complex plane and continiuous ? Is this really true in empty topological space ? probably I have a misunderstanding properties of zero ring

Answer 1

The point is that there only exists one function $\emptyset \rightarrow A$ for any set $A$, namely the empty function. If now $A$ happens to the set of real numbers $\mathbb{R}$ (or any other topological space), then this single function is continuous and as the set of continuous function from a topological space to the reals forms a ring, it is isomorphic to the zero ring (any ring with one element is isomorphic to the zero ring). There is no way to talk about any other function including the Riemann zeta function here though. These elements do not lie in that ring.