# compactness

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**Paracompact space**— In mathematics, a paracompact space is a topological space in which every open cover admits a locally finite open refinement. Paracompact spaces are sometimes also required to be Hausdorff. Paracompact spaces were introduced by Dieudonné (1944).… …42

**Maximum likelihood**— In statistics, maximum likelihood estimation (MLE) is a method of estimating the parameters of a statistical model. When applied to a data set and given a statistical model, maximum likelihood estimation provides estimates for the model s… …43

**Eberlein–Šmulian theorem**— In the mathematical field of functional analysis, the Eberlein–Šmulian theorem is a result relating three different kinds of weak compactness in a Banach space. The three kinds of compactness for a subset A of a topological space are: *… …44

**De Bruijn–Erdős theorem (graph theory)**— This article is about coloring infinite graphs. For the number of lines determined by a finite set of points, see De Bruijn–Erdős theorem (incidence geometry). In graph theory, the De Bruijn–Erdős theorem, proved by Nicolaas Govert de Bruijn and… …45

**Metric space**— In mathematics, a metric space is a set where a notion of distance (called a metric) between elements of the set is defined. The metric space which most closely corresponds to our intuitive understanding of space is the 3 dimensional Euclidean… …46

**Compact (disambiguation)**— Compact may mean: * Compact (newspaper), a broadsheet quality newspaper printed in a tabloid format. * Compact (soap opera), a 1960s British soap opera. * a diplomatic contract or covenant among parties, sometimes known as a pact, treaty, or an… …47

**Heine–Borel theorem**— In the topology of metric spaces the Heine–Borel theorem, named after Eduard Heine and Émile Borel, states:For a subset S of Euclidean space R n , the following two statements are equivalent: * S is closed and bounded *every open cover of S has a …48

**Weakly compact cardinal**— In mathematics, a weakly compact cardinal is a certain kind of cardinal number introduced by harvtxt|Erdös|Tarski|1961; weakly compact cardinals are large cardinals, meaning that their existence can neither be proven nor disproven from the… …49

**Löwenheim–Skolem theorem**— In mathematical logic, the Löwenheim–Skolem theorem, named for Leopold Löwenheim and Thoralf Skolem, states that if a countable first order theory has an infinite model, then for every infinite cardinal number κ it has a model of size κ. The… …50

**Finite intersection property**— In general topology, the finite intersection property is a property of a collection of subsets of a set X . A collection has this property if the intersection over any finite subcollection of the collection is nonempty.DefinitionLet X be a set… …