# Contact point VS boundary point

What is the difference between a contact point and a boundary point on a metric space ?

In topology and mathematics in general, the boundary of a subset S of a topological space X is the set of points which can be approached both from S and from the outside of S. More precisely, it 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.

In mathematics, an adherent point (also closure point or point of closure or contact point)[1] of a subset A of a topological space X, is a point x in X such that every open set containing x contains at least one point of A. A point x is an adherent point for A if and only if x is in the closure of A.

• Where have you come across these terms? It would be useful to work with your definitions of contact and boundary point. – Guido A. Oct 26 '18 at 10:24
• adherent point = contact point – Jim Art Oct 26 '18 at 10:49
• That's ok, but can you write up the definitions for these? It will make the difference clear, and if not, we can help you figure it out. – Guido A. Oct 26 '18 at 10:55

For example the boundary of the interval $$(0,1)$$ is $$\{0,1\}$$ but the closure is $$[0,1]$$.
For a definable topological space $$X$$ and a subset $$A\subseteq X$$ let $$cl(A)$$ denote the closure of $$A$$ and $$bd(A)$$ its boundary. Then $$bd(A)=cl(A)\cap cl(X\setminus A)$$.