# Countability Properties Preserved under Projection Mapping

## Theorem

Let $\sequence {\struct {S_\alpha, \tau_\alpha} }$ be a sequence of topological spaces.

Let $\ds \struct {S, \tau} = \prod \struct {S_\alpha, \tau_\alpha}$ be the product space of $\sequence {\struct {S_\alpha, \tau_\alpha} }$.

Let $\pr_\alpha: \struct {S, \tau} \to \struct {S_\alpha, \tau_\alpha}$ denote the projection on the $\alpha$ coordinate.

Then $\pr_\alpha$ preserves the following countability properties.

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

## 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:

are preserved under projections.

$\blacksquare$

## Sources

- 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.:
*Counterexamples in Topology*(2nd ed.) ... (previous) ... (next): Part $\text I$: Basic Definitions: Section $3$: Compactness: Invariance Properties