Important ideas that we discussed in this section include the following.
Let \(X\) be a metric space and \(A\) a subset of \(X\text{.}\)
A point \(x \in X\) is a boundary point of \(A\) if every neighborhood of \(x\) contains a point in \(A\) and a point in \(X \setminus A\text{.}\)
A point \(x\) is a limit point of \(A\) if every neighborhood of \(x\) contains a point in \(A\) different from \(x\text{.}\)
A point \(a \in A\) is an isolated point of \(A\) if there is a neighborhood \(N\) of \(a\) such that \(N \cap A = \{a\}\text{.}\)
Boundary points and limit points don’t need to be in the set \(A\text{,}\) whereas an isolated point of \(A\) must be in \(A\text{.}\) In \(A = (0,1) \cup \{2\}\) as a subset of \(=(\R, d_E)\text{,}\)\(0\) is a boundary point but not an isolated point while \(2\) is a boundary point but not a limit point. Also, \(0.5\) is a limit point but neither a boundary or isolated point. With \(A\) as subset of \(\R\) with the discrete metric, every point of \(A\) is an isolated point but no point in \(\R\) is a boundary point or a limit point of \(A\text{.}\) So even though every boundary point is either a limit point or an isolated point, the three concepts are different.
A subset \(A\) of a metric space \(X\) is closed if \(X \setminus A\) is an open set.
Any intersection of closed sets is closed while finite unions of closed sets are closed.
A function \(f\) from a metric space \(X\) to a metric space \(Y\) is continuous if \(f^{-1}(C)\) is a closed set in \(X\) whenever \(C\) is a closed set in \(Y\text{.}\)
Let \(X\) be a metric space, let \(A\) be a subset of \(X\text{,}\) and let \(x\) be a limit point of \(A\text{.}\) Then there is a sequence \((a_n)\) in \(A\) that converges to \(x\text{.}\)
Let \(X\) be a metric space, let \(A\) be a subset of \(X\text{,}\) and let \(x\) be a boundary point of \(A\text{.}\) Then there are sequences \((x_n)\) in \(X \setminus A\) and \((a_n)\) in \(A\) that converge to \(x\text{.}\)
The boundary of a subset \(A\) of a metric space \(X\) is the set of boundary points of \(A\text{.}\)
A subset \(A\) of a metric space \(X\) is closed if and only if \(A\) contains all of its limit points. Similarly, \(A\) is closed if and only if \(A\) contains all of its boundary points.
The set of all limit points of a subset \(A\) of a metric space \(X\) is denoted by \(A'\text{.}\) The closure of \(A\) is the set \(\overline{A} = A \cup A'\text{.}\) The closure of \(A\) is the smallest closed set in \(X\) that contains \(A\text{.}\)
A subset \(A\) of a metric space \(X\) is closed if and only if \(\lim a_n\) is in \(A\) whenever \((a_n)\) is a convergent sequence in \(A\text{.}\)
