Without specifying a precise answer to the question, I am surprised that there has been so little emphasis on continuity as the motivating concept for topology - topological spaces seem to me to have been designed, so to speak, to capture the notion of continuity in as much generality as seemed possible at the time, and particularly in non-metric contexts - and incidentally clarifying some proofs by throwing away the metric structure.
What can we recover from the epsilon-delta formulation of continuity if we don't allow measurement? It is possible that this question is more readily answered by reference to closed sets than to open ones.
Clearly the concept then takes off in all sorts of directions, where intuitions motivated by metrics are confounded (as mine was initially with the Zariski Topology).