For a normed space, the neighborhood of a point, is the open ball centered at where The closed ball, with is also a neighborhood. The usual notation (in this glossary) is and context dictates whether it is open or closed. This extends to the neighborhood of a set by taking the union; equivalently,
In integer programming, the neighborhood could mean integer-valued neighbors of the form In a combinatorial program, where variables are binary, integer-valued neighbors comprise all members of In this case the neighborhood is defined relative to some subset of binary variables reachable by some operation that depends on the problem. This is what is generally meant by a neighborhood in heuristic search in general, and simulated annealing or tabu search in particular, where a move is defined in terms of neighbors. In that context, a neighborhood could be as simple as complementing the value of one variable, as deletions or additions in a knapsack problem, or it could be more complex, like a -Opt neighbor of a TSP tour.