Topology can say two points are near without ever measuring distance
Open sets began as a generalisation of open intervals on the number line, the stretch between two numbers without its endpoints. Topologists realised that a well-chosen collection of such sets can capture closeness, continuity and compactness on its own, with no ruler anywhere in sight.
In a space with a distance, a set is open if every point in it comes with a small buffer: all points closer than some amount, which may depend on the point, also belong to the set. On the real line, distance is the absolute difference of two numbers, so the points within 1 of zero form the interval from minus 1 to 1, and within 0.5 they form the interval from minus 0.5 to 0.5. Shrinking that margin gives ever better approximations to zero, which suggests describing closeness through a family of sets rather than through the metric itself.
The general definition drops distance entirely. A topology on a set is a chosen family of subsets, the open sets, that contains the empty set and the whole set and is closed under any unions and under finite intersections. Those rules allow extremes. In the discrete topology every subset is open, and in the indiscrete topology only the empty set and the whole space are. If the whole real line were the only set used to gauge closeness to zero, every number would count as equally near it.
Usually, though, open sets are picked to mimic the nearness of a metric space. Two points are topologically distinguishable if some open set around one leaves out the other, and in that sense one can ask whether points or subsets are near each other. Continuity, connectedness and compactness, first defined with distances, can all be phrased this way, so topological spaces generalise metric ones.
Manifolds are the most familiar example of topology without distance: near every point they resemble an open region of Euclidean space, yet generally carry no built-in metric. Stranger topologies serve elsewhere, such as the Zariski topology at the foundations of algebraic geometry and scheme theory.
Source: Open set
