📂Topology

Limit Points and Convergence in Topological Spaces, Image Sets

Definition ¹

Let’s assume a topological space $\left( X , \mathscr{T} \right)$ is given.

1. If for $A \subset X$ any open set $O$ containing $x$ satisfies $O \cap ( A \setminus \left\{ x \right\} ) \ne \emptyset$, then $x$ is called a limit point of $A$, and the set of all limit points of $A$ is called the derived set of $A$.
2. A sequence $\left\{ x_{n} \right\}$ in $X$ converges to $x$ if for any open set $O$ containing $x$, there exists $n_{0} \in \mathbb{N}$ that satisfies the following: $$ n \ge n_{0} \implies x_{n} \in O $$

Explanation

Note that we do not specifically define divergence for sequences that do not converge.

Even in a topological space, limit points can still be defined, and there seems to be little difference in words. Nothing has changed from the definition in a metric space, but the conceptual difference is significant because although ‘all open sets’ were considered in a metric space, it felt like narrowing down the interval, whereas in a topological space, supposedly, all kinds of open sets must be considered.

That $A$ is a closed set in $X$ is equivalent to $ A ' \subset A$, which can be easily proved from the definitions of closed sets and limit points. More cleanly, it can be represented by $A = \overline{A}$ as well because of $\overline{A} = A \cup a '$. Especially, in metric spaces, it is expressed as the following theorem.

Theorem

In a metric space $(X,d)$, let’s say $K \subset X$.

• [1]: There exists a sequence of distinct points of $K$ that converges to $x \in X$, which is a limit point of $K$.
• [2]: If $K$ is a closed set in $X$, then all converging sequences of $K$ converge to a point of $K$.