No, it is not. Here is an example: [E. K. van Douwen, An anti-Hausdorff Fréchet space in which convergent sequences have unique limits, Top. Appl. Volume 51, Issue 2, 11 June 1993, Pages 147-158][1]


  [1]: https://doi.org/10.1016/0166-8641(93)90147-6