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Complete metric space, DYK Fact #166
 algebraic topology Posted: 15:12 Friday 30 March 2007 Renaissance Group: Friends Posts: 143 Member No.: 14 Joined: 20 Feb 2007 Did you know? A complete metric space is a metric space in which every Cauchy sequence converges.
Th.V
Posted: 07:52 Saturday 14 April 2007

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 QUOTE (algebraic topology @ 15:12 Friday 30 March 2007) Did you know? A complete metric space is a metric space in which every Cauchy sequence converges.

Complete metric space: A metric space, in which every Cauchy sequence is convergent, is a complete metric space. Did you know? Examples include the real numbers with the usual metric, the complex numbers, finite-dimensional real and complex vector spaces, (and also the space of square-integrable functions on the unit interval L²([0,1]), and the p-adic numbers). Let us notice however that Q is not a complete metric space, this leads thus to study an extension of Q to a completed space of Q.
 algebraic topology Posted: 23:04 Saturday 14 April 2007 Renaissance Group: Friends Posts: 143 Member No.: 14 Joined: 20 Feb 2007 Did you know? The completeness of the real numbers is often formulated in terms of the least-upper-bound property: every set of real numbers that is bounded above has a real least upper bound. It turns out that this is equivalent to the condition that every real Cauchy sequence is convergent.
 JaneFairfax Posted: 15:29 Wednesday 08 December 2010 The Enlightenment Group: Moderators Posts: 673 Member No.: 20 Joined: 03 Mar 2007 Let X be a metric space with metric d. Given a function f : X → X, suppose there exists a non-negative constant K < 1 such that for all x, y in X, d (f(x), f(y)) ≤ K d(x, y). (K can be 0, in which case f is constant.) The function f is called a contraction mapping on X. Did you know? Banach’s fixed-point theorem states that if X is a complete metric space, any contraction mapping f on X has a unique fixed point, i.e. there exists exactly one point p ∊ X such that f(p) = p.
 shygeorge Posted: 17:28 Thursday 09 December 2010 The Enlightenment Group: Moderators Posts: 534 Member No.: 161 Joined: 11 Dec 2009 Did you know? Stefan Banach (1892–1945) was a Polish mathematician.
 JaneFairfax Posted: 19:33 Thursday 09 December 2010 The Enlightenment Group: Moderators Posts: 673 Member No.: 20 Joined: 03 Mar 2007 Did you know? Baire’s theorem for metric spaces states that if X is a complete metric space and $\left\{A_n:n\in\mathbb{Z}^+\right\}$ is a countable collection of nowhere dense subsets of X, $X\setminus\bigcup_{n\,=\,1}^\infty{A_n}$ is everywhere dense in X. (A subset of a metric space X is nowhere dense iff the interior of its closure is empty; it is everywhere dense iff its closure is the whole space X.) Did you also know? René Baire (1874–1932) was a French mathematician.
 algebraic topology Posted: 14:18 Saturday 11 December 2010 Renaissance Group: Friends Posts: 143 Member No.: 14 Joined: 20 Feb 2007 Did you know? Any metric space can be “completed”. In a metric space X with metric d, let us say that two Cauchy sequences $\left(a_n\right)_{n\,=\,1}^\infty$ and $\left(b_n\right)_{n\,=\,1}^\infty$ are equivalent iff $\lim_{n\,\to\,\infty}d(a_n,\,b_n)=0$. This is quite clearly an equivalence relation. Denote the collection of all the equivalence classes by [X]. If $\left[\left(a_n\right)_{n\,=\,1}^\infty\right]$, $\left[\left(b_n\right)_{n\,=\,1}^\infty\right]$ ∊ [X], it can be shown that $\lim_{n\,\to\,\infty}d(a_n,\,b_n)$ exists. Moreover, the mapping $[d]:[X]\times[X]\to\mathbb{R}^+\cup\{0\}$ defined by $[d]\left(\left[\left(a_n\right)_{n\,=\,1}^\infty\right]\,\left[\left(b_n\right)_{n\,=\,1}^\infty\right]\right)=d(a_n,\,b_n)$ can be shown to be well defined and is a metric on [X]. ([X], [d]), then, is a complete metric space containing a homeomorphic copy of (X, d).
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