Note that without a separation axiom such as $T_1$, the result fails to hold. For example, any compact space is paracompact, and any fully normal space is normal, so any non-normal compact space is a paracompact space that’s not fully normal.

The Hausdorff condition from statement 2 cannot be dropped, again because there are compact $T_1$ spaces that are not normal, such as an infinite set with the cofinite topology.

References

A. H. Stone, Paracompactness and product spaces, Bull. Amer. Math. Soc. Volume 54, Number 10 (1948), 977-982. (Euclid)

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