Preimage of Open Sets forms Basis if Continuous Mappings Separate Points from Closed Sets

Theorem
Let $X$ be a topological space.

Let $\family {Y_i}_{i \mathop \in I}$ be an indexed family of topological spaces for some indexing set $I$.

Let $\family {f_i : X \to Y_i}_{i \mathop \in I}$ be an indexed family of continuous mappings.

Let $\family {f_i}_{i \mathop \in I}$ separate points from closed sets.

Then:
 * the set $\set{f_i^{-1} \sqbrk V : i \in I, V \text{ is open in } Y_i}$ is a basis for $X$

Proof
Let $U \subseteq X$ be open in $X$.

Let $x \in U$.

By definition of closed subset:
 * $X \setminus U$ is closed is $X$

By definition of mappings separating points from closed sets:
 * $\exists i \in I : \map {f_i} x \notin f_i \sqbrk {X \setminus U}^-$