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Dec 27, 2020 at 21:52 comment added Jochen Glueck Indeed, it is true in $T_3$-spaces. For instance, the Handbook of Analysis and its Foundations mentioned in Renan Maneli Mezabarba's answer states the result in $T_3$-spaces (in Section 17.15). I also had a look at the book Quasi-uniform spaces that you suggested: while I could not find exactly the same result there, the corollary to Proposition 3.15 seems to be at least related. Thank you for the reference!
Dec 26, 2020 at 9:10 history answered Martin Väth CC BY-SA 4.0