Countability Properties Preserved under Projection Mapping

Theorem
Let $\left \langle {\left({X_\alpha, \vartheta_\alpha}\right)}\right \rangle$ be a sequence of topological spaces.

Let $\displaystyle \left({X, \vartheta}\right) = \prod \left({X_\alpha, \vartheta_\alpha}\right)$ be the product space of $\left \langle {\left({X_\alpha, \vartheta_\alpha}\right)}\right \rangle$.

Let $\operatorname{pr}_\alpha: \left({X, \vartheta}\right) \to \left({X_\alpha, \vartheta_\alpha}\right)$ be the projection on the $\alpha$ coordinate.

Then $\operatorname{pr}_\alpha$ preserves the following countability properties.

That is, if $\left({X, \vartheta}\right)$ has one of the following properties, then each of $\left({X_\alpha, \vartheta_\alpha}\right)$ has the same property.


 * Separability


 * First-Countability


 * Second-Countability

Proof
First note that Projection from Product Topology is Continuous.

It follows from Separability is a Continuous Invariant that separability is preserved under projections.

Next note that Projection from Product Topology is Open.

It follows from Countability Axioms Preserved under Open Continuous Surjections that:
 * First-Countability
 * Second-Countability

are preserved under projections.