There was a similar thread on the neighbour forum StackExchange on sufficient conditions for a topological space to be completely regular $T_{3^1/_2}$.
Please, let me know any known condition(s) that a topological space is regular $T_3$. Any known approach or a standard strategy for showing that a topological space is regular would be very welcome and helpful answer too.
Thank you in advance!
EDIT: The topology I have in mind is a sequential ($\implies$ $k$-space, in the sense of Engelking) Lusin ($\implies$ Souslin, in the sense of Fernique) space topology defined via the Kantorovich-Vulih-Pinsker-Kisyński recipe; see e.g. Engelking's book Subsection 1.7.18-21 and references therein for the aforementioned K-V-P-K recipe.
UPDATE: Sufficient conditions for a topological space to be regular $T_3$ include:
-Compact Hausdorff spaces (admit a unique compatible uniformity $\implies T_{4}$),
-Uniform (e.g. metrizable $\implies T_6$) spaces,
-Topological groups (e.g topological vector spaces $\implies T_{3^1/_2}$),
-Arbitrary subspaces of a regular space,
-Arbitrary products of regular spaces.
I will keep updating the list for similar answers (as Todd's answer below). I thank everybody for their time and valuable help.