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The real numbers form a [[metric space]] with the usual distance as metric. As a [[topological space]], a subset A Euclidean space of fixed finite dimension ''n'' also forms a [[metric space]] with the Euclidean distance as metric. As a [[topological space]], the s

...em that states the existence and uniqueness of a fixed-point in a complete metric space.

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In [[mathematics]], a '''Cauchy sequence''' is a [[sequence]] in a [[metric space]] with the property that elements in that sequence ''cluster'' together mor ...ences may be convergent or not. This leads to the notion of a ''[[complete metric space]]'' as one in which every Cauchy sequence converges to a point of the space

The extended non-negative real exponent associated to any metric space where the Hausdorff measure changes from ∞ to 0.

...other forms throughout mathematics, and is encountered in the theory of [[metric space]]s in topology, the theory of [[normed vector space]]s in functional analys ...uality is ''assumed'' as one of the axioms for a metric space. Formally, a metric space is a set <math>X</math> equipped with a distance function <math>d: X \times

...ric space consists of two components, a set and a metric in that set. In a metric space, the metric replaces the Euclidean distance as a notion of "distance" betwe Every [[simple graph|simple]] [[graph]] can be viewed as a metric space (in more than one way). Thus formally the theory of simple graphs can be co

...''. The ''[[Baire category theorem]]'' states that a non-empty [[complete metric space]] is of second category.

A subset of a metric space with the property that for any positive radius it is coveted by a finite un

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...ch known as [[functional analysis]], a '''Hilbert space''' is a [[complete metric space|complete]] [[inner product space]]. As such, it is automatically also a [[B

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As a mathematical term, '''geometry''' refers to either the spatial ([[metric space|metric]]) properties of a given space or, more specifically in [[differenti

===Metric space=== In a [[metric space]] (''X'',''d''), a limit point of a set ''S'' may be defined as a point ''x

...Baire category theorem''' states that a non-[[empty set|empty]] [[complete metric space]] is a [[second category space]]: that is, it is not a [[countability|count

{{r|Complete metric space}}

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===Function on a metric space=== A function ''f'' from a [[metric space]] <math>(X,d)</math> to another metric space <math>(Y,e)</math> is ''continuous'' at a point <math>x_0 \in X</math> if f