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The word "bounded" makes no sense in a general topological space without a corresponding metric. Boundary is a distinct concept; for example, a circle (not to be confused with a disk) in isolation is a boundaryless bounded set, while the half plane is unbounded yet has a boundary. A bounded set is not necessarily a closed set and vice
The underlying idea is the following: a bounded set can be covered by a single ball of some radius. Sometimes several balls of a smaller radius can also cover the set. A compact set in fact can be covered by finitely many balls of arbitrary small radius, because it is totally bounded. So one could ask: what is the smallest radius that allows to ...
Bounded poset, a partially ordered set that has both a greatest and a least element; Bounded set, a set that is finite in some sense Bounded function, a function or sequence whose possible values form a bounded set; Bounded set (topological vector space), a set in which every neighborhood of the zero vector can be inflated to include the set
Database connections are finite and expensive and can take a disproportionately long time to create relative to the operations performed on them. It is inefficient for an application to create, use, and close a database connection whenever it needs to update a database. Connection pooling is a technique designed to alleviate this problem. A ...
The collection of all bounded sets on a topological vector space is called the von Neumann bornology or the (canonical) bornology of .. A base or fundamental system of bounded sets of is a set of bounded subsets of such that every bounded subset of is a subset of some . [1] The set of all bounded subsets of trivially forms a fundamental system of bounded sets of .
In topology and related branches of mathematics, total-boundedness is a generalization of compactness for circumstances in which a set is not necessarily closed.A totally bounded set can be covered by finitely many subsets of every fixed “size” (where the meaning of “size” depends on the structure of the ambient space).
Download as PDF; Printable version ... a uniformly bounded family of functions is a family of ... where is an arbitrary set and is the set of real or complex ...
If is the real line, or -dimensional Euclidean space, then a function has compact support if and only if it has bounded support, since a subset of is compact if and only if it is closed and bounded. For example, the function f : R → R {\displaystyle f:\mathbb {R} \to \mathbb {R} } defined above is a continuous function with compact support ...