There are many other equivalent ways to define a topological space: in other words the concepts of neighbourhood, or that of open or closed sets can be reconstructed from other starting points and satisfy the correct axioms.
Metric spaces embody a metric, a precise notion of distance between points.
Any local field has a topology native to it, and this can be extended to vector spaces over that field.
Any set can be given the cocountable topology, in which a set is defined as open if it is either empty or its complement is countable. When the set is uncountable, this topology serves as a counterexample in many situations.