To me, the concept of an open set is a distillation and abstraction of the properties of open intervals (on the real line) that are critical to defining and working with a continuous function. In my opinion students should never be introduced to the abstract concepts of topology (notably, the concepts of open sets and compactness) unless they have already mastered analysis of functions on the real line and finite-dimensional vector spaces and understand thoroughly the role of open sets and compactness in those settings.
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