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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.

A mathematical structure A is said to be closed under an operation + if, whenever a and b are both elements of A, then so is a+b. A mathematical object taken together with ...

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.

A function has compact support if it is zero outside of a compact set. Alternatively, one can say that a function has compact support if its support is a compact set. For ...

A complete metric space is a metric space in which every Cauchy sequence is convergent. Examples include the real numbers with the usual metric, the complex numbers, ...