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These are two examples in which both the subset and the whole set are infinite, and the subset has the same cardinality (the concept that corresponds to size, that is, the number of elements, of a finite set) as the whole; such cases can run counter to one's initial intuition.
As an example, let be the set of all finite subsets of natural numbers and consider the partially ordered set obtained by taking all sets from together with the set of integers and the set of positive real numbers +, ordered by subset inclusion as above.
In topology and related areas of mathematics, a subset A of a topological space X is said to be dense in X if every point of X either belongs to A or else is arbitrarily "close" to a member of A — for instance, the rational numbers are a dense subset of the real numbers because every real number either is a rational number or has a rational number arbitrarily close to it (see Diophantine ...
The examples use the poset (({,,}),) consisting of the set of all subsets of a three-element set {,,}, ordered by set inclusion (see Fig. 1). a is related to b when a ≤ b . This does not imply that b is also related to a , because the relation need not be symmetric .
In mathematics, a subset of a given set is closed under an operation of the larger set if performing that operation on members of the subset always produces a member of that subset. For example, the natural numbers are closed under addition, but not under subtraction: 1 − 2 is not a natural number, although both 1 and 2 are.
A partition of a set X is a set of non-empty subsets of X such that every element x in X is in exactly one of these subsets [2] (i.e., the subsets are nonempty mutually disjoint sets). Equivalently, a family of sets P is a partition of X if and only if all of the following conditions hold: [3]
For instance, had been declared as a subset of , with the sets and not necessarily related to each other in any way, then would likely mean instead of . If it is needed then unless indicated otherwise, it should be assumed that X {\displaystyle X} denotes the universe set , which means that all sets that are used in the formula are subsets of X ...
Given a topological space (,), the following statements are equivalent: . a set is closed in . = is an open subset of (,); that is, . is equal to its closure in .; contains all of its limit points.