level-set-of-a-continuous-function-must-be-closed

[utterances-bot]

A Universe of Sorts

https://bollu.github.io/level-set-of-a-continuous-function-must-be-closed.html

[oxidist]

Even simpler: {y} is closed in a Hausdorff topology, so f^(-1) ({y}) is closed since f is continuous. (In fact your sequence reasoning only works assuming Hausdorffness, otherwise you'd use nets)