ThmDex – An index of mathematical definitions, results, and conjectures.

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Result R4470 on D529: Map inverse image
Subresult of R31: Inverse image of union is union of inverse images

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 $n \in \mathbb{N}$

$\square$