Characterization of Paracompactness in T3 Space

Theorem
Let $T = \struct{X, \tau}$ be a $T_3$ space.


 * $(1)\quad T$ is paracompact
 * $(2)\quad $every open cover of $T$ has a locally finite refinement
 * $(3)\quad $every open cover of $T$ has a closed locally finite refinement
 * $(4)\quad $every open cover of $T$ is even
 * $(5)\quad $every open cover of $T$ has an open $\sigma$-discrete refinement
 * $(6)\quad $every open cover of $T$ has an open $\sigma$-locally finite refinement
 * $(7)\quad T$ is fully $T_4$