There was a similar thread on the neighbour forum [StackExchange](https://math.stackexchange.com/questions/114053/are-topological-vector-spaces-completely-regular) 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.



Sufficient conditions for a topological space to be regular $T_3$
The list of sufficient conditions in the question neatly avoids addressing the real issue I think. From the comments I see that it is regularity of a very specific topology that you are after: Jakubowski's $S$-topology on Skorokhod space. None of the conditions that you have will help you very much in that case.
The reason is that proving normality or (complete) regularity requires fairly detailed knowledge of the nature of the open sets. The Kisynski construction yields a topology but with very little information on what the closed sets look like, other than that they are closed under taking limits.
Hausdorffness of the $S$-topology is forced by having a point-separating family of continuous functions.
For regularity (to prove it or disprove it) you need to know what arbitrary open sets look like; that would require a very detailed study of the convergence used (I must admit that I did not have the time for that).


To see how difficult it may be to deduce properties of the topology from properties of the convergence have a gander at [this example](https://doi.org/10.1016/0166-8641(93)90147-6): a $T\_1$-space that is Frechet-Urysohn, with unique sequential limits but that is very far from Hausdorff.


In short: if you want to (dis)prove regularity of the $S$-topology you'll have to get your hands dirty.


