GF Simmons, Introduction to Topology and Modern Analysis Section 11, Pg 68-69
Let $X$ be a metric space and $A$ a subset of $X$. A point in $X$ is called a boundary point of $A$ if each open sphere centered on the point intersects both $A$ and A', and the boundary of $A$ is the set of all boundary points. This concept possesses the following properties:
(1) The boundary of $A$ equals A \cap A';
(2) The boundary of $A$ is a closed set;
(3) $A$ is closed $\iff$ it contains boundary
The first property is wrong I suppose? Else all boundary sets will be empty sets. Any idea what can be a replacement to that property? For example, did the author actually intend to say that
(1) The boundary of $A$ equals \bar{A} \cap A'
where $\bar{A}$ means closure of $A$.