Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [complete-spaces]

A metric space is complete if, in it, any Cauchy sequence is convergent.

Intuitively, a space is complete if there are no “points missing” from it, as far as limits of sequences are concerned. For instance, the set of rational numbers is not complete, because e.g. $\sqrt{2}$ is “missing” from it, even though one can construct a sequence of rational numbers that converges to it, which is necessarily a Cauchy sequence. It is always possible to “fill all the holes”, leading to the completion of a given space.

When working on a complete space, one can determine that a sequence converges by proving that it is a Cauchy sequence, thereby avoiding the need of actually determining its limit. This is very useful in Analysis.

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Prove/Disprove Metric Space is Complete

Let $X=\left\{(x,1/x) : x\in(0,\infty) \right\}$ be a metric space with $$d((x,1/x),(y,1/y)) = \sqrt{(x-y)^2+(1/x-1/y)^2}.$$ Prove whether or not the space is complete. I can't find a counter-example, and also my intuition is that it is indeed…
Yoni
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Is $(A,\rho)$ complete?

A is a set of all null sequences $x=(\alpha_n)$ of complex numbers C. $\rho=sup_i|\alpha_i-\beta_i|$ is a metric on A such that $x=(\alpha_n)$ and $y=(\beta_n)$. Assume C to be complete under the usual metric. Is $(A,\rho)$ complete? I know that if…
Sepia Glen
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(verification) Set of Cluster points of a sequence $x_k$ is closed.

Claim Set of Cluster(Accumulation) points of a sequence $x_k$ is closed. Proof A set is closed if and only if its accumulation(limit) point in that set. So we can change the claim into "set of cluster points of a sequence $x_k$ holds the limit…
Daschin
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Is a complete metric space a continuum?

By definition, a continuum is a compact connected metric space. Is it correct to say that a complete metric space is always compact connected? In other words, can I say that a complete metric space is a continuum? Thanks in advance
Roberto Dias Algarte
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$\mathbb{Q}[\sqrt{2}]$ is not a complete space

I am trying to prove that $\mathbb{Q}[\sqrt{2}]=\{a+b\sqrt{2}:a,b\in\mathbb{Q}\}$ is not a complete space (with the standard metric). For that purpose, I am looking for a Cauchy sequence in $\mathbb{Q}[\sqrt{2}]$ which is not convergent in…
JN_2605
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show that the metric space = { + : , ∈ ℚ} with usual metric is incomplete

Let (,d) be a metric space with = { + : , ∈ ℚ} and d( , ) = |x − y|. Show that (,d) is incomplete I haven't been able to find any Cauchy sequence that is divergent in X other than this one: x1= 1 and xn+1= 2*(1 +xn)/(2 +xn), for n≥2 ,which is a…
Cookiemaster
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Completeness with equivalent norm

Suppose I have a subset $C$ of a Banach space $X$ which is complete respect to a given (norm induced by a) metric $d$. Let also suppose that it exist a subset $S\subset C$ which is a metric space respect to a metric $d'$ and that it holds: $$A…
aleio1
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Why is the following space not complete

why is (C[0,1],||.||4) not a complete space? Since it is closed, is it because of the norm 4? I know it should be complete if for every sequence its limit is in the space.
user404056
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Another way to prove completedness in a metric space

Let $X=\mathbb{N}$ and $g(m,n)=0$ if $m=n$, and $g(m,n)=1+\frac{1}{m+n}$ if $m\neq n$. I've proved g is a metric. Is the structure (X,g) complete? sadly I'm not familiar enough with those problems to intuitively deduce it is or not complete. To…
José Osorio
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