The advantage of defining open/closed sets using sequence. I have learned that in a first-countable topological space, open and closed sets can be defined using sequences. Moreover, the interior or closure of a set can be defined in similar ways.
My question is, what is the advantage of using sequences in testing whether a set is open or closed? Are there any particular problems where sequences are more useful than neighborhoods?
 A: Sequences are more familiar from metric spaces and it's often easier to say which sequences converge than to describe the open or closed sets. As a simple example, the product topology on the Cantor set $\{ 0, 1 \}^{\mathbb{N}}$ is exactly the topology of pointwise convergence, meaning a sequence $f_n$ of functions $f_n : \mathbb{N} \to \{ 0, 1 \}$ converges iff each of its values $f_n(m)$ converges, for all $m \in \mathbb{N}$.
Meanwhile the product topology is the coarsest topology with respect to which all evaluation maps $f_n \mapsto f_n(m)$ are continuous; this is really saying the same thing but you might find the open sets of the product topology (generated by cylinder sets) abstract and harder to think about.
For non-sequential spaces the use of sequences can be replaced by nets or filters, but then you'd have to learn about convergence of nets and filters. This is a fine thing to do but many people have not done it (I don't know anything about nets myself) whereas lots more people have taken courses in real analysis at some point. So you can rely on more people understanding an argument involving sequences.
