Let $\Omega$ be a compact Hausdorff space in $\mathbb{C}^n$. Let $\sigma_\Omega$ be the Borel sigma algebra on $\Omega$. Let $\zeta: \Omega\longrightarrow\partial \mathbb{D}$ be a non constant continuous function. Let $\sigma_{\partial \mathbb{D}}$ be the Borel sigma algebra on $\partial \mathbb{D}$(Unit circle on the complex plane). Now consider the sigma algebra $\sigma_\zeta=\{{\zeta}^{-1}(A): \;A\in \sigma_{\partial \mathbb{D}}\}\subset \sigma_\Omega$.
I wanted to know if this new sigma algebra $\sigma_\zeta$ will contain the singleton elements? If not, will there be some additional conditions on $\zeta$ such that $\sigma_\zeta$ will definitely contain the singletons.
The $\sigma$-algebra $\sigma_\zeta $ contains all singletons if and only if it is injective. If it is injective, every singleton from $\Omega$ is the preimage of a singleton. If it is not injective, there must be $\omega$ and $\omega'$ with $\zeta(\omega)=\zeta'(\omega)$. Then for every subset $E$ of the codomain, $\zeta^{-1}(E)$ contains either both $\omega$ and $\omega'$ or neither of them.