Definition 13.3.
Let \(X\) be a topological space, and let \(A\) be a subset of \(X\text{.}\) A limit point of \(A\) is a point \(x \in X\) such that every neighborhood of \(x\) contains a point in \(A\) different from \(x\text{.}\)
Section Limit Points and Sequences in Topological Spaces
Recall that we defined a limit point of a set \(A\) in a metric space \(X\) to be a point \(x \in X\) such that every neighborhood of \(x\) contains a point in \(A\) different from \(x\text{.}\) Since we have defined neighborhoods in topological spaces, we can make the same definition.
The set \(A'\) of limit points of \(A\) is called the derived set of \(A\text{.}\)
Activity 13.3.
Find the limit point(s) of the following sets
(a)
\(\{c,d\}\) in \((X, \tau)\) with \(X= \{a,b,c,d\}\) and \(\tau = \{\emptyset, \{a\}, \{b\}, \{a,b\}, X \}\)
(b)
\(\{a,b\}\) in the set \(X= \{a,b,c,d,e,f\}\) with topology
\begin{equation*}
\tau= \{\emptyset,\{b\}, \{a,b,c\},\{d,e,f\},\{b,d,e,f\}, X\}\text{.}
\end{equation*}
(c)
\(\{a,b\} \subset X\) where \(X = \{a,b,c\}\) in the discrete topology.
(d)
\(\{-1,0,1\} \subset \Z\) with \(\tau\) the topology on \(\Z\) with basis \(\{B(n)\}\text{,}\) where
\begin{equation*}
B(n) = \begin{cases}\{n\} \amp \text{ if \(n\) is odd } , \\ \{n-1,n,n+1\} \amp \text{ if \(n\) is even } . \end{cases}
\end{equation*}
(This topology is called the digital line topology and has applications in digital processing. That the collection \(\{B(n)\}\) is a basis for a topology on \(\Z\) is the topic of Exercise 11)
In metric spaces, a set is closed if and only if it contains all of its limit points. So the corresponding result in topological spaces should be no surprise.
Theorem 13.4.
Let \(C\) be a subset of a topological space \(X\text{,}\) and let \(C'\) be the set of limit points of \(C\text{.}\) Then \(C\) is closed if and only if \(C' \subseteq C\text{.}\)
Proof.
Let \(X\) be a topological space, and let \(C\) be a subset of \(X\text{.}\) First we assume that \(C\) is closed and show that \(C\) contains all of its limit points. Let \(x \in X\) be a limit point of \(C\text{.}\) We proceed by contradiction and assume that \(x \notin C\text{.}\) Then \(x \in X \setminus C\text{,}\) which is an open set. This means that there is a neighborhood (namely \(X \setminus C\)) of \(x\) that contains no points in \(C\text{,}\) which contradicts the fact that \(x\) is a limit point of \(C\text{.}\) We conclude that \(x \in C\) and \(C\) contains all of its limit points.
For the converse, assume that \(C\) contains all of its limit points. To show that \(C\) is closed, we prove that \(X \setminus C\) is open. We again proceed by contradiction and assume that \(X \setminus C\) is not open. Then there exists \(x \in X \setminus C\) such that no neighborhood of \(x\) is entirely contained in \(X \setminus C\text{.}\) This implies that every neighborhood of \(x\) contains a point in \(C\) and so \(x\) is a limit point of \(C\text{.}\) It follows that \(x \in C\text{,}\) contradicting the fact that \(x \in X \setminus C\text{.}\) We conclude that \(X \setminus C\) is open and \(C\) is closed.
In metric spaces we saw that limit point of a set is the limit of a sequence of points in the set. To explore this idea in topological spaces, we define sequences in the same way we did in metric spaces.
Definition 13.5.
A sequence in a topological space \(X\) is a function \(f: \Z^+\) to \(X\text{.}\)
We use the same notation and terminology related to sequences as we did in metric spaces: we will write \((x_n)\) to represent a sequence \(f\text{,}\) where \(x_n = f(n)\) for each \(n \in \Z^+\text{.}\) We can’t define convergence in a topological space using a metric, but we can use open sets. Recall that a sequence \((x_n)\) in a metric space \((X,d)\) converges to a point \(x\) in the space if, given \(\epsilon \gt 0\) there exists a positive integer \(N\) such that \(d(x_n,x) \lt \epsilon\) for all \(n \geq N\text{.}\) In other words, every open ball centered at \(x\) contains all of the entries of the sequence past a certain point. We can replace open balls with open sets and make a similar definition of convergence in topological spaces.
Definition 13.6.
A sequence \((x_n)\) in a topological space \(X\) converges to the point \(x \in X\) if, for each open set \(O\) that contains \(x\) there exists a positive integer \(N\) such that \(x_n \in O\) for all \(n \geq N\text{.}\)
If a sequence \((x_n)\) converges to a point \(x\text{,}\) we call \(x\) a limit of the sequence \((x_n)\text{.}\)
Activity 13.4.
In metric spaces, limits of sequences are unique. We may wonder if the same result is true in topological spaces. Consider the topological space \((X,\tau)\text{,}\) where \(X = \{a, b, c\}\) and \(\tau = \{\emptyset, \{c\}, \{a, c\}, \{b, c\}, X\}\text{.}\) Find all limits of all constant sequences in \(X\text{.}\)
The result of Activity 13.4 is that sequences do not behave in topological spaces as we would expect them to. Consequently, sequences do not play the same important role in topological spaces as they do in metric spaces. However, the concept of limit point is important, as are the notions of boundary and closure in topological spaces.
