Near sets are by definition pairs of nonempty sets that have non void intersection either spatially or descriptively. A collection of near sets can be viewed in a number of different ways. Here are two examples:

Signature collection of near sets (call it N1): A collection of nonempty sets that are close to a given set sigX (the signature of the collection). A representative member of N1 is a pair (sigX, Y) with sigX close to Y, either spatially or descriptively.

Characteristic collection of near sets (call it Nk): A collection of pairs of sets (Xi, Yi), 1

Near sets appear in the study of connected structures in any physical space.

A connected structure contains a path between any pair of points (vertexes) in the structure. Here are a couple of examples:

rainbow: Between any pair of photons of the same wavelength, there is a path between the photons. In that case, photons in the same path are near each other and photons with close wavelengths are near sets. By attaching line segments to the vertexes (locations of path photons) to form a skeleton representing a photon path, and assuming that we move forwards and backwards along the constructed skeleton, we then have the basis for a cyclic abelian group (G, +).

For simplicity, the binary operation on G can be addition modulo 2. Also make the assumption that each skeleton edge in G is a multiple of a single edge in the path. So, if we move along n edges in the path, the reversal of that movement gives us -n edges, the additive inverse of the n edges. The same of n edges will either be 1 or 0. The group identity of G is 0. We also know that + (mod 2) is both commutive and associative. A pair of cyclic groups (G, +) and (G',+) are spatially near sets, provided G and G' have elements in common. And the cyclic groups G and G' are descriptively near sets, provided G and G' have matching descriptions. For example, if we select path-length as a feature of a group and G, G' have the same path length, then G and G' are descriptively near sets of photons.

garden: Between any pair of plants in a garden row, there is a path between the plants. Plants in the same garden row are connected, since the plants belong to intersection subpaths (the subpath A containing one plant in the same row in a subrow B overlapping subrow A. A pair of neighbouring garden rows are both connected and constitute close (near) sets, since rows have a common border. In a manner similar to the rainbow cyclic abelian group, we can also construct a garden cyclic abelian group.

Perhaps the followers of this thread can suggest other examples of connectedness in the physical world that are not so obvious.

In terms of the algebra on connected structures a physical space,