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A (symmetrical) boundary set of radius r and center x_0 is the set of all points x such that |x-x_0|=r. Let x_0 be the origin. In R^1, the boundary set is then the pair of ...

A mathematical object (such as a set or function) is said to bounded if it possesses a bound, i.e., a value which all members of the set, functions, etc., are less than.

A set is said to be bounded from above if it has an upper bound. Consider the real numbers with their usual order. Then for any set M subset= R, the supremum supM exists (in ...

A set is said to be bounded from below if it has a lower bound. Consider the real numbers with their usual order. Then for any set M subset= R, the infimum infM exists (in R) ...

For every positive integer n, there exists a square in the plane with exactly n lattice points in its interior. This was extended by Schinzel and Kulikowski to all plane ...

A subset of a topological space is called clopen if it is both closed and open.

The closed ball with center x and radius r is defined by B_r(x)={y:|y-x|<=r}.

A map f between topological spaces that maps closed sets to closed sets. If f is bijective, then f is closed <==>f is open <==>f^(-1) is continuous, where f^(-1) denotes the ...

A set U has compact closure if its set closure is compact. Typically, compact closure is equivalent to the condition that U is bounded.

A subset of a topological space which is compact with respect to the relative topology.