Questions tagged [metric-spaces]

Metric spaces are sets on which a metric is defined. A metric is a generalization of the concept of "distance" in the Euclidean sense. Metric spaces arise as a special case of the more general notion of a topological space.

10,254 questions
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Closed spaces in a metric space.

Let $(X,d)$ a metric space. Let $F$ and $A$ two subsets of $X$ such that $A\cap F=\emptyset$ and $F$ is closed. Suppose that for any converging sequence $\{u_{n}\}_{n\in \mathbb{N}}\subset A$, we have ...
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Continuity preserves limits in metric spaces [on hold]

Let M be a metric space, $\mathit{f}: M \rightarrow \mathbb{R}$ continuous and a convergent succession $x_{n}$ in M, proof that: $$\lim_{n \to \infty} f(x_{n}) = f(\lim_{n \to \infty}x_{n})$$
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What are the correct options in the following questions

I know that 2, 4 are correct. Are option 1,3 are correct?
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Are Two Metric Spaces Equivalent?

Are the following metrics equivalent on $\Bbb R$? $d(x,y)=|x-y|$ and $d'(x, y)=|\tan^{-1}(x) - \tan^{-1}(y)|$.
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Prove the following set is open in $\mathbb R$ with the standard metric.

Let $x_k = (\sin k, \arctan k, k^3)$, $k \in \mathbb Z_+$. Prove (by the open ball definition) that $V = \mathbb R^3\setminus\{x_k: k \in \mathbb Z_+\}$ is open in $\mathbb R$ with the standard metric....
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A metric space in which $3^\infty=2^\infty=0$

I want a space containing all the positive integers in which $3^nx+3^n-2^n\to0$ as $n\to\infty$ Perhaps paradoxically, numbers not factorisable by $2,3$ would be a sufficient set for me (in case that ...
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Boundary of a set in metric space

I'm trying to find the boundary of a set in order to check whether or not the set is open or closed on the metric I'm given. While I understand the latter, I'm not sure how to properly understand the ...
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recurrent sequence in a compact space

please how to solve this question Let $E$ a compact metric space and a function $f:E\to E$ such that $$\forall x,y \in E, d(f(x),f(y))\geq d(x,y)$$ Let $a\in E$ and a sequence $(f^n(a))$ , ...
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Complete space of functions with $f(a)=f(b)$?

Consider M is space of continuous functions (on $[a,b]$) with condition: $f(a)=f(b)$ . Is it complete metric space with $\mu(f,g) =\max\underset{x\in [a,b]}{|(f(x)-g(x))|}$? In my opinion it's true. ...
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Homeomorphism between $([0,1]^2, \delta )$ and $[0,1]^2$ with standard euclidean metric

I am stuck as to how I find the homeomorphism described above. $\delta$ is here described to be the metric $$\delta((m_1,n_1), (m_2,n_2))= max\{d_M(m_1,m_2),d_N(n_1,n_2)\}$$ a metric on $M\times N$ ...
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Convergence of measure on clopen basis implies the existence of a limit measure?

Let $X$ be a compact metrizable space. Let $B$ be basis for the topology of $X$. Assume that all sets in $B$ are clopen. Let $(\mu_n)_{n=1}^{\infty}$ be sequence of Borel regular probability measures ...
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How do I visualize a set in metric space?

If I'm given a metric, say the discrete metric $\text d_0(x,y):=\begin{cases} 0, & \text{if }\vec x= \vec y\\ 1, & \text{if } \vec x \neq \vec y\ \end{cases}$ and want to visualize a set ...
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A sequence of continuously differentiable functions with non-differentiable limit

Working in the metric space $C[a,b]$, the set of continuous functions $f:[a,b]\rightarrow[a,b]$, with the supremum metric, I need to demonstrate that $C^1[a,b]$, the subset of continuously ...
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Separability of a product metric space

I am trying to prove the following: 'If $(X_1,d_1)$ and $(X_2,d_2)$ are separable metric spaces (that is, they have a countable dense subset), then the product metric space $X_1 \times X_2$ is ...
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Equality of certain distances in a normed vector subspace

Let $E$ be a normed vector space and $F$ a vector subspace of $E$. If $y \in F$, $x \in E$ and $0 < a \in \mathbb R$, prove that $d(y + ax, F) = ad(x,F)$. I've tried to write down the definitions ...
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Metric on Lorentzian manifold

Since the signature of a Lorentzian manifold (M,g) is (-,+,+,+), am I right to assume that the determinant of $g_{ij}$ with EACH coordinate system is $-1$?
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A set $\Omega$ is bounded if there exists $M > 0$ such that $|z| < M$ whenever $z \in \Omega$. The set $\Omega$ is contained in some large disc.

My complex analysis textbook says the following: A set $\Omega$ is bounded if there exists $M > 0$ such that $|z| < M$ whenever $z \in \Omega$. In other words, the set $\Omega$ is contained ...
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Proof of Schur's Theorem

On Pg. 123 of Schaum's Tensor Calculus: At an isotropic point of $R^n$ the Riemannian curvature is given by $$K=\frac{R_{abcd}}{g_{ac}g_{bd}-g_{ad}g_{bc}}=\frac{R_{abcd}}{G_{abcd}}$$ for any ...
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Define a metric on collection of compacts

Let $(\mathbb{R}^2,d)$ be a metric space, where $d$ is the Euclidean metric on $\mathbb{R}^2$. Let $\kappa$ denote the set of compact subsets of $R^2$. Which one of the following expression defines a ...
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Do these define metrics on $\mathbb{R}$?
Which of the following define a metric on $\mathbb{R}$? $$d_1(x,y) = \frac{\bigl||x|-|y|\bigr|} {1+|x||y|},$$ $$d_2(x,y) = \frac{\bigl||x|+|y|\bigr|} {1+|x||y|}.$$ I think that both option 1 and ...
Is $X$ totally bounded?
I came across the following claim in this post. Let $X$ be a metric space in which every infinite subset has a limit point. Then for every $\delta>0$ there exists $N_{\delta}\in \mathbb{N}$ and \$...