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A subset S of a partially ordered set P is called bounded if it has both an upper and a lower bound, or equivalently, if it is contained in an interval. Note that this is not just a property of the set S but also one of the set S as subset of P. A bounded poset P (that is, by itself, not as subset) is one that has a least element and a greatest ...
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 .
A subset S of a uniform space X is totally bounded if and only if, for any entourage E, there exists a finite cover of S by subsets of X each of whose Cartesian squares is a subset of E. (In other words, E replaces the "size" ε , and a subset is of size E if its Cartesian square is a subset of E .) [ 4 ]
The subset (1, 2) is a bounded interval, but it has no infimum or supremum in P, so it cannot be written in interval notation using elements of P. A poset is called locally finite if every bounded interval is finite.
Bornology originates from functional analysis.There are two natural ways of studying the problems of functional analysis: one way is to study notions related to topologies (vector topologies, continuous operators, open/compact subsets, etc.) and the other is to study notions related to boundedness [2] (vector bornologies, bounded operators, bounded subsets, etc.).
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 ...
But this is just the least element of the whole poset, if it has one, since the empty subset of a poset P is conventionally considered to be both bounded from above and from below, with every element of P being both an upper and lower bound of the empty subset. Other common names for the least element are bottom and zero (0).
X is closed and bounded (as a subset of any metric space whose restricted metric is d). The converse may fail for a non-Euclidean space; e.g. the real line equipped with the discrete metric is closed and bounded but not compact, as the collection of all singletons of the space is an open cover which admits no finite subcover. It is complete but ...