Sequence of Implications of Paracompactness Properties

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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:


Compact $\implies$ Countably Compact
$\Big\Downarrow$ $\Big\Downarrow$
Fully Normal $\implies$ Paracompact $\implies$ Countably Paracompact
$\Big\Downarrow$ $\Big\Downarrow$ $\Big\Downarrow$
Fully $T_4$ Metacompact $\implies$ Countably Metacompact
$\Big\Downarrow$
$T_4$


Proof

The relevant justifications are listed as follows:

$\blacksquare$


Sources