# Paracompactness is not always Preserved under Open Continuous Mapping

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## Theorem

Let $T_A = \struct {X_A, \tau_A}$ be a topological space which is paracompact.

Let $T_B = \struct {X_B, \tau_B}$ be another topological space.

Let $\phi: T_A \to T_B$ be a mapping which is both continuous and open.

Then it is not necessarily the case that $T_B$ is also paracompact.

## Proof

We also have:

Paracompact Space is Metacompact
Metacompact Space is Countably Metacompact

Hence the result.

$\blacksquare$