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A minimal element of a subset of some preordered set is defined dually as an element of that is not greater than any other element in . The notions of maximal and minimal elements are weaker than those of greatest element and least element which are also known, respectively, as maximum and minimum.
Let X be a topological space, and let a subset be negligible if it is of first category, that is, if it is a countable union of nowhere-dense sets (where a set is nowhere-dense if it is not dense in any open set). Then the negligible sets form a sigma-ideal. Let X be a directed set, and let a subset of X be negligible if it has an upper bound ...
The axiom of constructibility is a possible axiom for set theory in mathematics that asserts that every set is constructible.The axiom is usually written as V = L.The axiom, first investigated by Kurt Gödel, is inconsistent with the proposition that zero sharp exists and stronger large cardinal axioms (see list of large cardinal properties).
If there is a set that is a standard model of ZF, then the smallest such set is such a L κ. This set is called the minimal model of ZFC, and also satisfies the axiom of constructibility V=L. The downward Löwenheim–Skolem theorem implies that the minimal model (if it exists as a set) is a countable set.
In mathematics, the limit of a sequence of sets,, … (subsets of a common set ) is a set whose elements are determined by the sequence in either of two equivalent ways: (1) by upper and lower bounds on the sequence that converge monotonically to the same set (analogous to convergence of real-valued sequences) and (2) by convergence of a sequence of indicator functions which are themselves ...
In the mathematical field of general topology, a meagre set (also called a meager set or a set of first category) is a subset of a topological space that is small or negligible in a precise sense detailed below. A set that is not meagre is called nonmeagre, or of the second category. See below for definitions of other related terms.
In a totally ordered set the maximal element and the greatest element coincide; and it is also called maximum; in the case of function values it is also called the absolute maximum, to avoid confusion with a local maximum. [1] The dual terms are minimum and absolute minimum. Together they are called the absolute extrema. Similar conclusions ...
A set of real numbers (hollow and filled circles), a subset of (filled circles), and the infimum of . Note that for totally ordered finite sets, the infimum and the minimum are equal. A set of real numbers (blue circles), a set of upper bounds of (red diamond and circles), and the smallest such upper bound, that is, the supremum of (red diamond).