ThmDex – An index of mathematical definitions, results, and conjectures.
Result R4470 on D529: Map inverse image
Inverse image of countable union is union of inverse images
Formulation 0
Let $f : X \to Y$ be a D18: Map such that
(i) $B_n \subseteq Y$ is a D78: Subset of $Y$ for each $n \in \mathbb{N}$
Then \begin{equation} f^{-1} \left( \bigcup_{n \in \mathbb{N}} B_n \right) = \bigcup_{n \in \mathbb{N}} f^{-1}(B_n) \end{equation}
Proofs
Proof 0
Let $f : X \to Y$ be a D18: Map such that
(i) $B_n \subseteq Y$ is a D78: Subset of $Y$ for each $n \in \mathbb{N}$
This result is a particular case of R31: Inverse image of union is union of inverse images. $\square$