Topological space
More precisely, a topological space has a certain kind of set, called open sets. Open sets are important because they allow one to talk about points near another point, called a neighbourhood of the point. A neighbourhood of a point is simply an open set containing that point. If one did not have the concept of open sets, one cannot define neighbourhoods in a good way. If one tries to define a neighbourhood of a point as any set containing that point, it might just include that point and that point only, not any points near it, or points far away.