# Possible error about properties of boundary points in Simmons's Topology and Modern Analysis

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 $A$.

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If by $A'$ one means the complement of $A$, then the correct replacement is: $\partial A = \overline{A}\cap \overline{A'}$ ($\partial A$ is a common notation in geometry for the boundary). – William Mar 29 '12 at 6:48

It looks as if he meant $\overline{A} \cap \overline{A^\prime}$. Have a look here.