Completely agreed. Part of this might look like a shift in definitions/​vocabulary over time. Coming to topology from analysis, sequence felt like a natural way to interrogate limiting behavior. After a while though, it sort of became clear that thinking sequentially requires putting first-countability assumptions everywhere. Introducing nets did away with the need for this assumption and better captured what convergence ought to mean in a general topological spaces.
Completely agreed. Part of this might look like a shift in definitions/​vocabulary over time. Coming to topology from analysis, sequence felt like a natural way to interrogate limiting behavior. After a while though, it sort of became clear that thinking sequentially requires putting first-countability assumptions everywhere. Introducing nets did away with the need for this assumption and better captured what convergence ought to mean in a general topological spaces.