Sequence of Implications of Paracompactness Properties

Theorem
Let $P_1$ and $P_2$ be paracompactness properties and let:
 * $P_1 \implies P_2$

mean:
 * If a topological space $T$ satsifies property $P_1$, then $T$ also satisfies property $P_2$.

Then the following sequence of implications holds:

Proof
The relevant justifications are listed as follows:


 * Compact Space is Countably Compact.


 * Compact Space is Paracompact.


 * Fully Normal Space is Paracompact.


 * Countably Compact Space is Countably Paracompact.


 * Paracompact Space is Countably Paracompact.


 * Fully Normal Space is Fully $T_4$ by definition.


 * Fully $T_4$ Space is $T_4$.


 * Paracompact Space is Metacompact.


 * Metacompact Space is Countably Metacompact.


 * Countably Paracompact Space is Countably Metacompact.