View source for Sequential space

{{TEX|done}}{{MSC|54A20}}

A [[topological space]] $X$ such that if $A\subset X$ and $A\neq[A]$ (that is, the set $A$ is not closed), then there is a sequence $x_k$, $k=1,2,\dots,$ of points of $A$ that converges to a point of $[A]\setminus A$. If $x\in[A]\subset X$ always implies that there is a sequence $x_k$ of points from $A$ that converges to $x$, then $X$ is called a '''Fréchet–Urysohn space'''.



====Comments====
Sequential spaces form a coreflective subcategory (see [[Reflective subcategory|Reflective subcategory]]) of the category of all topological spaces; the coreflection is obtained by retopologizing an arbitrary space with the topology in which a subset is closed if and only if it is closed under limits of sequences (in the original topology). Spaces which satisfy the [[First axiom of countability|first axiom of countability]] are always sequential (indeed, they are Fréchet–Urysohn spaces), and the sequential spaces form the smallest coreflective subcategory containing all first-countable spaces. For this reason, many topological results which are traditionally proved for first-countable spaces can readily be extended to sequential spaces.

Every Fréchet–Urysohn space is a sequential space.

====References====
<table>
<TR><TD valign="top">[a1]</TD> <TD valign="top"> R. Engelking, "General topology" , Heldermann (1989)</TD></TR>
<TR><TD valign="top">[a2]</TD> <TD valign="top"> Jerzy Kąkol, Wiesław Kubiś, Manuel López-Pellicer, "Descriptive topology in selected topics of functional analysis" Springer (2011) {{ISBN|978-1-4614-0528-3}} {{ZBL|1231.46002}}</TD></TR>
</table>