# Paracompactness is not always Preserved under Open Continuous Mapping

Jump to navigation
Jump to search

## 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:

Hence the result.

$\blacksquare$

## Sources

- 1970: Lynn Arthur Steen and J. Arthur Seebach, Jr.:
*Counterexamples in Topology*... (previous) ... (next): $\text{I}: \ \S 3$: Invariance Properties