Boundary

Definition: Boundary

Let $(X,\tau )$ be a topological space and let $S$ be a subset of $X$.

The boundary of $S$ is the set of all points $x\in X$ such that every neighbourhood of $x$ contains at least one point of $S$ and at least one point of its complement $X\setminus S$.

NOTATION

$$\partial S{\partial }_{X}S{\mathrm{Bd}}_{X}S$$

Definition: Boundary Point

Let $(X,\tau )$ be a topological space and let $S$ be a subset of $X$.

A point $p\in X$ is a boundary point of $S$ iff it belongs to the boundary of $S$.

$$p\in \partial S$$