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A set such as {{,,}} is a singleton as it contains a single element (which itself is a set, but not a singleton). A set is a singleton if and only if its cardinality is 1. In von Neumann's set-theoretic construction of the natural numbers, the number 1 is defined as the singleton {}.
A locally compact Hausdorff space has small inductive dimension 0 if and only if it is totally disconnected. Every totally disconnected compact metric space is homeomorphic to a subset of a countable product of discrete spaces. It is in general not true that every open set in a totally disconnected space is also closed.
This article lists mathematical properties and laws of sets, involving the set-theoretic operations of union, intersection, and complementation and the relations of set equality and set inclusion. It also provides systematic procedures for evaluating expressions, and performing calculations, involving these operations and relations.
An example of this is R 3 = R × R × R, with R again the set of real numbers, [1] and more generally R n. The n-ary Cartesian power of a set X is isomorphic to the space of functions from an n-element set to X. As a special case, the 0-ary Cartesian power of X may be taken to be a singleton set, corresponding to the empty function with codomain X.
Every totally bounded set is bounded. A subset of the real line, or more generally of finite-dimensional Euclidean space, is totally bounded if and only if it is bounded. [5] [3] The unit ball in a Hilbert space, or more generally in a Banach space, is totally bounded (in the norm topology) if and only if the space has finite dimension.
The fibers of are that line and all the straight lines parallel to it, which form a partition of the plane . More generally, if f {\displaystyle f} is a linear map from some linear vector space X {\displaystyle X} to some other linear space Y {\displaystyle Y} , the fibers of f {\displaystyle f} are affine subspaces of X {\displaystyle X ...
A class diagram exemplifying the singleton pattern.. In object-oriented programming, the singleton pattern is a software design pattern that restricts the instantiation of a class to a singular instance.
In fact, a measure on the real line is a Dirac measure for some point if and only if the support of is the singleton set {}. Consequently, Dirac measure on the real line is the unique measure with zero variance (provided that the measure has variance at all).