At least for me, the first time I learned about a metric space, was to discuss when sequences converge. I am not a math-history buff, but the concept of metric seems to stem from the need to formalize the concept of convergence. However, everything about of convergence depends only on the topology the metric generates. Hence one only needs to understand the open sets.

Topological spaces generalize metric spaces in the sense that every metric space gives rise to one and all concepts convergence are captured by this topological space. Even more, the category of metric spaces and continuous maps sits inside the category of topological spaces as a full subcategory (this is just saying that a map between metric spaces is continuous if and only if the preimage of an open is open). However, you may object, and justifiably, that you can embed metric spaces into several categories, so, why is topological spaces the right generalization?

There are many "practical answers", for instance, the wide range of examples of abstract spaces which are not metric spaces, e.g., Spec(R) of a ring. However, the core of the matter is that topological spaces correctly axiomatize the notion of convergence. What I mean by this is, a topological space is completely determined by the convergence of the ultrafilters on its underlying set. This is seem most notably for compact Hausdorff spaces; a topological space is compact Hausdorff if and only iff every ultrafilter has a unique limit, and in fact, compact Hausdorff spaces are precisely the algebras of the ultrafilter monad. However, we are interested in non-compact examples, since we study unbounded metric spaces for example- but even here we can have ultrafilters with no limit- so, a complete generalization should take this into account. Furthermore, the space Spec(R) is very often non-Hasudorff, which means, ultrafilters which have a limit point, may have more than one. So, to understand convergence is to understand the set of limits of each ultrafilter. If X is a space and BX is its set of ultrafilters, we get a map BX->P(X) which sends each ultrafilter to its set of limit points (possibly empty). This corresponds to a relation $R \subset X \times BX$, which made be seen as a map BX->X in the bicategory of sets and relations. More precisely, we get a "relational algebra" for the ultrafilter monad. The converse is true as well: the category of topological spaces is equivalent to the category of relational algebras for the ultrafilter monad. This a theorem of Barr. The upshot is, there is a bijection between topologies on a set and "convergence systems for ultrafilters" on that set.

Anyhow, this probably goes way beyond what you can explain to most undergraduates.

At least for me, the first time I learned about a metric space, was to discuss when sequences converge. I am not a math-history buff, but the concept of metric seems to stem from the need to formalize the concept of convergence. However, everything about of convergence depends only on the topology the metric generates. Hence one only needs to understand the open sets.

Topological spaces generalize metric spaces in the sense that every metric space gives rise to one and all concepts of convergence are captured by this topological space. Even more, the category of metric spaces and continuous maps sits inside the category of topological spaces as a full subcategory (this is just saying that a map between metric spaces is continuous if and only if the preimage of an open is open). However, you may object, and justifiably, that you can embed metric spaces into several categories, so, why is topological spaces the right generalization?

There are many "practical answers", for instance, the wide range of examples of abstract spaces which are not metric spaces, e.g., Spec(R) of a ring. However, the core of the matter is that topological spaces correctly axiomatize the notion of convergence. What I mean by this is, a topological space is completely determined by the convergence of the ultrafilters on its underlying set. This is seem most notably for compact Hausdorff spaces; a topological space is compact Hausdorff if and only iff every ultrafilter has a unique limit, and in fact, compact Hausdorff spaces are precisely the algebras of the ultrafilter monad. However, we are interested in non-compact examples, since we study unbounded metric spaces for example- but even here we can have ultrafilters with no limit- so, a complete generalization should take this into account. Furthermore, the space Spec(R) is very often non-Hasudorff, which means, ultrafilters which have a limit point, may have more than one. So, to understand convergence is to understand the set of limits of each ultrafilter. If X is a space and BX is its set of ultrafilters, we get a map BX->P(X) which sends each ultrafilter to its set of limit points (possibly empty). This corresponds to a relation $R \subset X \times BX$, which made be seen as a map BX->X in the bicategory of sets and relations. More precisely, we get a "relational algebra" for the ultrafilter monad. The converse is true as well: the category of topological spaces is equivalent to the category of relational algebras for the ultrafilter monad. This a theorem of Barr. The upshot is, there is a bijection between topologies on a set and "convergence systems for ultrafilters" on that set.

Anyhow, this probably goes way beyond what you can explain to most undergraduates.