Paracompactness is not always Preserved under Open Continuous Mapping

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 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

• 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