# 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.

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### Can we classify all topological spaces where separability of every subspace implies the the topological space is metrizable?

$(X, \tau)$be a topological space and $(Y, \tau_Y)$ be it's topological subspace. I know if $(X,\tau)$ is separable and metrizable then $(Y,\tau_Y)$ is separable. Suppose $(Y, \tau_Y)$ is separable ...
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### Show that $(C[0,1],d_\infty) \rightarrow (C[0,1],d_1)$ is continuous

$d_\infty = \sup |f(x) - g(x)|) \$ and $\ d_1 = \int_{i=1}^n |f(x) - g(x)|$. I have seen how does the "other direction" work, I mean $(C[0,1],d_\infty \rightarrow (C[0,1],d_1)$ and I also ...
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### Some question about $\ell_p$ and metric $d_p$ with $p\in (0,1)$

Hi I trying to see if $d_p=\sum_{k=1}^\infty|x_k-y_k|^p$ with $p\in(0,1)$ define a metric on $\ell_p$ my idea 1.first I need to see $\ell_p$ with $p\in (0,1)$ is not empty and $d_p$ Is good define. So ...
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### What are some examples of "Jordan spaces" which are *not* homeomorphic to a subspace of $\Bbb R^n$ (with the Euclidean topology)?

Note: This question has been substantially revised, see the edit history for earlier versions. So far, the only examples of metric spaces which I have seen in topology books are Euclidean $n$-space, ...
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### What's the relation between (strict) convexity of unit balls and shortest distance paths in $l_p$ metric?

I'm reading the book Geometry of Quantum States by Bengtsson and Zyczkowski. They have a brief discussion on $l_p$ norms. Depending on circumstances, different choices of p may be particularly ...
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### Let $A$ be a subset of a metric space $(X,d)$… [closed]

(c). Let $A$ be a subset of a metric space $(X,d)$. Prove that (i). $A$ is open if and only if $A=int(A)$. $$\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad [4\text{ marks}]$$ (ii). $A$ ...
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### How to show that $A_t := \bigg\{u \in \mathbb{R}^n \mid u_1 = 0 \text{ and } \sum_{i=1}^n \lvert u_{i+1}-u_i \rvert \le t \bigg\}$ is closed?

I am having trouble with the following exercise: For $t > 0$ consider the set $$A_t := \bigg\{u \in \mathbb{R}^n \mid u_1 = 0 \text{ and } \sum_{i=1}^n \lvert u_{i+1}-u_i \rvert \le t \bigg\}.$$ ...
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### Is the interval $[a,b]$ in $\Bbb{R}$ compact with the discrete metric?

I'm wondering whether or not a closed, bounded interval $[a,b]$ in $\Bbb{R}$ is compact with the discrete metric? I tried to show it wasn't by taking the set of singletons in $[a,b]$ as an open cover,...
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### Close functions have close points

Suposse that $f,g:\mathbb{R}\to\mathbb{R}$ are $C^1$ and $$\int_0^1{|f-g|^2+|f'-g'|^2~dx}<\varepsilon$$ I would like to know if it is possible to say something about the distance between $f(x_0)$ ...
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### Do the curvature properties of exotic spheres result in necessary new techniques for calculating arc length?

It was made clear to me in the post What answer does one get from integrating a Riemannian metric on a sphere for a great circle – does the metric’s non-flatness affect the answer? that a great circle ...
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### Portmanteau theorem with lower semi-continuous and bounded from below functions

I'm trying to prove this version of Portmanteau theorem. Could you verify if my attempt is fine? Let $(X, d)$ be a metric space and $\mathcal P(X)$ the space of all Borel probability measures on $X$. ...
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### Approximate a lower semi-continuous and bounded from below function by an increasing sequence $(f_n)$ of $n$-Lipschitz continuous functions

I'm trying to prove this result. Could you verify if my attempt is fine? Let $(X, d)$ be a metric space. Let $f:X \to \mathbb R \cup \{+\infty\}$ be a lower semi-continuous and bounded from below. ...
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### $d_{\mathbb{H}}(p,q)=|\log(pq;rs)|$.

I'd like some assistance in demonstrating $d_{\mathbb{H}}(p,q)=|\log(pq;rs)|$ where $\mathbb{H}$ is a Poincare Upper Half-Plane, and $(pq;rs)$ is a cross ratio of $p,q,r,s$. ($p,q$ are on a geodesic ...
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