I'm having trouble showing a certian function is open and can be extended.
Let $\Omega$ be a completely regular topological space and $A=C_b(\Omega)$ the space of all complex-values bounded continuous functions on $\Omega$. For every $\lambda \in \Omega$, the consider the functional $\phi_{\lambda}:$ $A\to \mathbb C $ that sends $f\to f(\lambda)$ (which belongs to the characters of $A$ denoted bt $\Phi (A)$) then I proved that the map$$\delta:\Omega\to \Phi (A)$$ $$\lambda \to \phi_{\lambda}$$ is injective, continuous and $\operatorname{Im}\delta$ is dense. I need help with the following:
- Why the following sets are form a bases of $\Omega$ $$\{\lambda\in\Omega:|f_i(\lambda)-f_i(\lambda_{0})|<\epsilon,i=1,2,\dots,n\},$$ where $\lambda_{0}\in\Omega$,$f_1,f_2,\dots,f_n\in A$?
- Why the sets in 1. gets mapped to an open set under $\delta$?
- Why every complex bounded continuous function on $\Omega$ can be uniquely extended to $\Phi (A)$?
For part 3), I'm thinking to consider every such function as a functional and use Hahn-Banach Theorem.
I need your help please any ideas.