In my understanding, a sequentially closed subset $A\subset X$ of a topological space $X$ is one that contains every sequential limit point of itself, whereas the sequential closure of $A$ is defined as $[A]_{seq}=\{x\in X: \exists (x_n)\in A^{\mathbb{N}}: x_n\to x\}$.
I think that the sequential closure need not be sequentially closed, because assuming to the contrary that every sequential closure was sequentially closed, I can construct the contradiction that every sequential space is a Frechet-Urysohn space as follows:
Let $X$ be a sequential space. We must show $\overline{A}=[A]_{seq}$.
"$\overline{A}\subset[A]_{seq}$": $[A]_{seq}$ is sequentially closed, thus closed since $X$ is sequential. Also $A\subset[A]_{seq}$ trivially. Hence $\overline{A}=\bigcap_{A\subset F\subset X closed}F \subset [A]_{seq}$.
"$\overline{A}\supset[A]_{seq}$": This is true for every topological space (every limit of a sequence is part of the closure)
Am I right? If yes, isn't the notation exceptionally inconvenient?
Best Answer
Yes, you're right: Fréchet-Uryson spaces are precisely the spaces in which the sequential closure is the same as the ordinary closure. For a specific example you can use the Arens space, which is discussed in Dan Ma’s Topology Blog: the sequential closure of the set of isolated points of the Arens space is not closed.
It's true that the sequential closure operator is not a true closure operator in the usual sense of the term; this is a minor nuisance, and one might wish for a better term, but I hardly think that it qualifies as exceptionally inconvenient.
Added: To turn the sequential closure operator into a true closure operator, you have to iterate it, possibly transfinitely. That is, if $\operatorname{scl}$ denotes the sequential closure, define
$$\operatorname{scl}^\eta A=\begin{cases} A,&\text{if }\eta=0\\\\ \operatorname{scl}\bigcup_{\xi<\eta}\operatorname{scl}^\xi A,&\text{if }\eta>0\;. \end{cases}$$
There is always an ordinal $\eta$ such that $\operatorname{scl}^\xi A=\operatorname{scl}^\eta A$ for all $\xi\ge\eta$; if we define $\operatorname{scl}^*A=\operatorname{scl}^\eta A,$ then $\operatorname{scl}^*$ is a true closure operator, and a space $X$ is sequential iff $\operatorname{scl}_X^*$ is identical to the ordinary closure.