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Conversely, a set which is not bounded is called unbounded. 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.
If a set is both closed and unbounded, then it is a club set. Closed proper classes are also of interest (every proper class of ordinals is unbounded in the class of all ordinals). For example, the set of all countable limit ordinals is a club set with respect to the first uncountable ordinal ; but it is not a club set with respect to any ...
A set that is not bounded is called unbounded. Bounded sets are a natural way to define locally convex polar topologies on the vector spaces in a dual pair , as the polar set of a bounded set is an absolutely convex and absorbing set .
For example, the feasible set defined by the constraint set {x ≥ 0, y ≥ 0} is unbounded because in some directions there is no limit on how far one can go and still be in the feasible region. In contrast, the feasible set formed by the constraint set { x ≥ 0, y ≥ 0, x + 2 y ≤ 4} is bounded because the extent of movement in any ...
The set of all bounded sequences forms the sequence space. [ citation needed ] The definition of boundedness can be generalized to functions f : X → Y {\displaystyle f:X\rightarrow Y} taking values in a more general space Y {\displaystyle Y} by requiring that the image f ( X ) {\displaystyle f(X)} is a bounded set in Y {\displaystyle Y} .
In mathematical analysis, limit superior and limit inferior are important tools for studying sequences of real numbers.Since the supremum and infimum of an unbounded set of real numbers may not exist (the reals are not a complete lattice), it is convenient to consider sequences in the affinely extended real number system: we add the positive and negative infinities to the real line to give the ...
Noticing any new changes to your skin, especially if a new growth pops up, only naturally sets off a series of internal alarm bells: Is that a mole, a wart, a melanoma?If that little bump on your ...
A well-ordered set also has the least-upper-bound property, and the empty subset has also a least upper bound: the minimum of the whole set. An example of a set that lacks the least-upper-bound property is , the set of rational numbers.