Thread: Sequences and closed sets. View Single Post
 Mentor P: 18,278 Hi Frederik! Every closed set of a topological space is closed under limits of sequences! It's the converse that's not true. That is: there are sets which are not closed but which are still closed under limits of sequences. For example, take the cocountable topology. Let X be a set and set $$\mathcal{T}=\{A\subseteq X~\vert~X\setminus A~\text{is countable}\}\cup\{\emptyset\}$$ Every convergent sequence in this topology is (eventually) a constant sequence. Thus all sets are closed under limits of sequences. But not all sets are closed, of course. Some terminology: a set that is closed under limits of sequences is called sequentially closed. A topological space where closed is equivalent with sequentially closed, is called a sequential space. As is well-known, all first-countable spaces are sequential.