Just by havin the notion of neighbourhood, concepts such as limit and continuity can be defined without the need to specify a precise numerical value to the distance between two points with a metric.
As an example. consider the orthogonal group, which is also naturally a topological space. That group does not usually have a notion of distance defined for it by default. However, we can still talk about certain properties of it, e.g. that the orthogonal group is compact, and that the orthogonal group has two connected components.
Topology is a branch of mathematics that deals with the properties of space that are preserved under continuous transformations, such as stretching, twisting, crumpling, and bending, but not tearing or gluing. It focuses on the concepts of structure, continuity, and convergence, and is often described as "rubber-sheet geometry" because of its emphasis on the flexible and qualitative aspects of geometric forms.
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