# 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:

 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:

$\blacksquare$