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A bounded set is not necessarily a closed set and vice versa. For example, a subset S of a 2-dimensional real space R 2 constrained by two parabolic curves x 2 + 1 and x 2 - 1 defined in a Cartesian coordinate system is closed by the curves but not bounded (so unbounded).
An interval is said to be left-closed if it has a minimum element or is left-unbounded, right-closed if it has a maximum or is right unbounded; it is simply closed if it is both left-closed and right closed. So, the closed intervals coincide with the closed sets in that topology.
The Cantor set is an unusual closed set in the sense that it consists entirely of boundary points and is nowhere dense. Singleton points (and thus finite sets) are closed in T 1 spaces and Hausdorff spaces. The set of integers is an infinite and unbounded closed set in the real numbers.
Many important linear operators which fail to be bounded turn out to be closed, such as the derivative and a large class of differential operators. Let X, Y be two Banach spaces. A linear operator A : D(A) ⊆ X → Y is closed if for every sequence {x n} in D(A) converging to x in X such that Ax n → y ∈ Y as n → ∞ one has x ∈ D(A ...
A "closed universe" is necessarily a closed manifold. An "open universe" can be either a closed or open manifold. For example, in the Friedmann–Lemaître–Robertson–Walker (FLRW) model, the universe is considered to be without boundaries, in which case "compact universe" could describe a universe that is a closed manifold.
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 higher limit ordinal, since it is neither closed nor unbounded.
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 .
Intuitively, the graph of a bounded function stays within a horizontal band, while the graph of an unbounded function does not. In mathematics , a function f {\displaystyle f} defined on some set X {\displaystyle X} with real or complex values is called bounded if the set of its values is bounded .