Closure (topology) in the context of "Boundary (mathematics)"

In topology and mathematics in general, the boundary of a subset S of a topological space X is the set of points in the closure of S not belonging to the interior of S. An element of the boundary of S is called a boundary point of S. The term boundary operation refers to finding or taking the boundary of a set. Notations used for boundary of a set S include ${\displaystyle \operatorname {bd} (S),\operatorname {fr} (S),}$ and ${\displaystyle \partial S}$.

In mathematics, specifically in topology,the interior of a subset S of a topological space X is the union of all subsets of S that are open in X.A point that is in the interior of S is an interior point of S.The interior of S is the complement of the closure of the complement of S.In this sense interior and closure are dual notions.

The exterior of a set S is the complement of the closure of S; it consists of the points that are in neither the set nor its boundary.The interior, boundary, and exterior of a subset together partition the whole space into three blocks (or fewer when one or more of these is empty).

In the field of topology, a Fréchet–Urysohn space is a topological space ${\displaystyle X}$ with the property that for every subset ${\displaystyle S\subseteq X}$ the closure of ${\displaystyle S}$ in ${\displaystyle X}$ is identical to the sequential closure of ${\displaystyle S}$ in ${\displaystyle X.}$ Fréchet–Urysohn spaces are a special type of sequential space.

The property is named after Maurice Fréchet and Pavel Urysohn.