Paracompactness is not always Preserved under Open Continuous Mapping

Theorem
Let $I$ be an indexing set.

Let $\family {\struct {S_\alpha, \tau_\alpha} }_{\alpha \mathop \in I}$ be a family of topological spaces indexed by $I$.

Let $\displaystyle \struct {S, \tau} = \prod_{\alpha \mathop \in I} \struct {S_\alpha, \tau_\alpha}$ be the product space of $\family {\struct {S_\alpha, \tau_\alpha} }_{\alpha \mathop \in I}$.

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

If $\struct {S, \tau}$ is paracompact, then it is not always the case that each of $\struct {S_\alpha, \tau_\alpha}$ is also paracompact.

Proof
We have Open Continuous Image of Paracompact Space is not always Countably Metacompact.

We also have:
 * Paracompact Space is Metacompact
 * Metacompact Space is Countably Metacompact

Hence the result.