Open Continuous Image of Paracompact Space is not always Countably Metacompact

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 even the case that $T_B$ is countably metacompact, let alone paracompact.

Proof
Let $T_X = \struct {X, \tau}$ be a countable discrete space.

Let $T_Y = \struct {\set {0, 1}, \tau_0}$ be a Sierpiński space.

Let $T_A = T_X \times T_Y$ be the product space of $T_X$ and $T_Y$.

Then $T_A$ is paracompact from Product of Countable Discrete Space with Sierpiński Space is Paracompact.

Let $T_B = \struct {S, \tau_p}$ be a countable particular point space.

Then $T_B$ is not countably metacompact from Infinite Particular Point Space is not Countably Metacompact

Let $\phi: T_A \to T_B$ be a mapping defined as:
 * $\forall \tuple {x, y} \in T_A: \map \phi {x, y} = \begin {cases}

p & : y = 0 \\ x & : y = 1 \end {cases}$

Then $\phi$ is open and continuous as follows:

So we have constructed $\phi$ which is both open and continuous from a paracompact space to a space which is not countably metacompact.