Paracompactness is not always Preserved under Open Continuous Mapping

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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.

$\blacksquare$


Sources