Countability Properties Preserved under Projection Mapping

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Theorem

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

Let $\displaystyle \left({S, \tau}\right) = \prod \left({S_\alpha, \tau_\alpha}\right)$ be the product space of $\left \langle {\left({S_\alpha, \tau_\alpha}\right)}\right \rangle$.


Let $\operatorname{pr}_\alpha: \left({S, \tau}\right) \to \left({S_\alpha, \tau_\alpha}\right)$ be the projection on the $\alpha$ coordinate.

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

That is, if $\left({S, \tau}\right)$ has one of the following properties, then each of $\left({S_\alpha, \tau_\alpha}\right)$ has the same property.

Separability
First-Countability
Second-Countability


Proof

First note that Projection from Product Topology is Continuous.

It follows from Continuous Image of Separable Space is Separable that separability is preserved under projections.


Next note that Projection from Product Topology is Open.

It follows from Countability Axioms Preserved under Open Continuous Surjection that:

First-Countability
Second-Countability

are preserved under projections.

$\blacksquare$


Sources