Understanding two similar definitions: Fréchet-Urysohn space and sequential space Here are the definitions:

Fréchet-Urysohn space: A topological space $ X $ where for every $ A \subseteq X $ and every $ x \in \text{cl}(A) $, there exists a sequence $ (x_{n})_{n \in \mathbb{N}} $ in $ A $ converging to $ x $.
Sequential space: A topological space $ X $ where a set $ A \subseteq X $ is closed iff $ A $ contains the limit points of every sequence contained in it.

As the title explains, I would like to know the difference between them. Thanks for any help.
Consider the following operation on a subset $A$ of a space $X$, defining a new subset of $X$: $$\mbox{s-cl}(A) = \{ x \in X \mid \mbox{ there exists a sequence } (x_n)_n \mbox{ from } A \mbox{ such that } x_n \rightarrow x \}\mbox{.}$$ This set, the sequential closure of $A$, contains $A$ (take constant sequences) and in all spaces $X$ it will be a subset of the $\mbox{cl}(A)$, the closure of $A$ in $X$. 

We can define $\mbox{s-cl}^{0}(A) = A$ and for ordinals $\alpha &gt; 0$ we define $\mbox{s-cl}^\alpha(A) = \mbox{s-cl}(\cup_{\beta &lt; \alpha} \mbox{s-cl}^\beta(A))$, the so-called iterated sequential closure.  

A space is Fréchet-Urysohn when $\mbox{s-cl}(A) = \mbox{cl}(A)$ for all subsets $A$ of $X$, so the first iteration of the sequential closure is the closure.

A space is sequential if some iteration $\mbox{s-cl}^\alpha(A)$ equals the $\mbox{cl}(A)$, for all subsets $A$. 

So basically by taking sequence limits we can reach all points of the closure eventually in a sequential space, but in a Fréchet-Urysohn space we are done after one step already.

For more on the differences and the "canonical" example of a sequential non-Fréchet-Urysohn space (the Arens space), see this nice topology blog, and the links therein.
a space is a Sequential space  space iff every sequencially closed set is closed.But there can be closed sets which are not sequencially closed.
a space is frechet-urysohn if sequencial closure of a set and usual closure coincide