The second example of convergent sequences in topological spaces demonstrates that the terms topological space and convergence are so generally defined that a sequence can have more than one limit point.
If there are not "enough" open sets in a topology $\mathcal O$ of a topological space $(X,\mathcal O),$ for instance, $X$ has more points than $\mathcal O$ has open sets, then limits of sequences in such a space $X$ will be ambiguous.
On the other hand, the first example of convergent sequences in topological spaces shows that "to many" open sets in a topology make the study of convergent sequences trivial and therefore too obvious to be interesting.
Clever restrictions to the topology $\mathcal O$ of topological spaces, called separation axioms, ensure interesting properties of convergence, for instance, the uniqueness of limits known from analysis. This part of BookofProofs is dedicated to the separation axioms and the conclusions they allow.