Preservation ledger for the separation axioms, with $T_1$ conventions kept explicit

$T_0$, $T_1$, Hausdorffness, regularity, $T_3$, complete regularity, and Tychonoffness pass both to subspaces and to arbitrary products by $T_0$, $T_1$, $T_2$, regularity, $T_3$, complete regularity, and Tychonoffness are hereditary and $T_0$, $T_1$, $T_2$, regularity, $T_3$, complete regularity, and Tychonoffness are productive. The compound names retain their $T_1$ clauses: $T_3$ and Tychonoff are not alternative names for regularity and complete regularity.

Normality has a narrower positive result: it passes to closed subspaces by Every closed subspace of a normal space is normal. Complete normality is exactly hereditary normality by A space is completely normal if and only if every subspace is normal. Under countable choice the deleted Tychonoff plank refutes hereditary normality (Assuming countable choice, normality is not hereditary, even to open regular subspaces); under choice the lower-limit plane refutes productive normality (Assuming choice, normality is not productive: the normal lower-limit line has a nonnormal square). These failures do not alter the positive ledger above.