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. For questions about Riemannian metrics use the tag (riemannian-geometry) instead.
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Given a metric $d$, is continuity of $f$ absolutely essential for the composite function $f \circ d$ to be a metric (equivalent to $d$)? [closed]
Let $d$ be a metric on a nonempty set $X$, and let $f \colon [0, +\infty) \longrightarrow [0, +\infty)$ be a function such that (i) $f(0) = 0$, (ii) $f(r+s)\leq f(r) + f(s)$ for all $r, s \in [0, +\...
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Is there a maximum number of disjoint balls of fixed radius I can fit into a compact metric space?
Let $(X,d)$ be a compact metric space and fix $r>0$. By sequential compactness, one may not find an infinite number of disjoint $r$-balls (sets $B_r(x):=\{y \in X: d(x,y)<r\}$) in $X$ as this ...
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Connected Metric Spaces: Strategies
I am not really sure if my ideas in this topic are correct. Can anyone help me?
Finding the connected components of a metric space $X$.
Suppose there are two connected components $C_1, C_2$ of $X$. ...
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Convex combination of equidistant curves
Say we have three curves $\gamma, \delta, \varepsilon : \mathbb R \to \mathbb R^n$ such that the distances $\lVert \gamma(t) - \delta(t) \rVert$ and $\lVert \gamma(t) - \varepsilon(t) \rVert$ are ...
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Can any open set in $\mathbb{R}^d$ be countably union of closed sets
I've already know that $\{B(x,r):x\in\mathbb{Q}^d,r\in \mathbb{Q}\} $ is an countable base of $\mathbb{R}^d$. Intuitively, I wonder that can an open set $\Omega\subset \mathbb{R}^d$ be countably union ...
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Is a metric/distance not a measure?
A metric (https://en.wikipedia.org/wiki/Metric_space) or a distance (as in premetric) takes two elements of a set and maps the pair to a real number (or maybe even a complex number). It also might ...
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Finishing the proof of the triangle inequality of Hausdorff metric
currently I am trying to show that a certain type of Hausdorff metric satisfies the following triangle equality and I am stuck.
Setup:
Take $(X,d)$ as metric space.
Denote by $C(X)$ the set of closed ...
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How to understand the Sobolev space defined by completion.
In page 16 of this book,
the author state:
For $1\leq p <\infty$, consider the normed space of all smooth functions $\phi \in \mathbb{R}^n$ such that
$$
\|\phi\|_{1,p} = \|\phi\|_p + \|\nabla \phi\...
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Composition of asymmetric contraction mappings [closed]
Let $(M,d)$ and $(N,q)$ be metric spaces.
The operator $T:M\longrightarrow N$ is contractive in the sense that $q(T(m_1),T(m_2)) \leq c d(m_1, m_2)$ for some $c\in [0,1)$.
Similarly, the operator $J:N\...
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Convex cocompact representation of finitely generated groups
Let $\mathbb{H}^n$ be the hyperboloid model for hyperbolic space and $\text{Isom}(\mathbb H^n) = PO(n,1)$.
Let $\rho: \Gamma \rightarrow PO(n,1)$ be a representation of finitely generated group $\...
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Is my understanding of the definition of a metric space correct?
The definition of metric space that I am using is as follows:
Let $X$ be a nonempty set. A function $d:X\times X\to \Bbb R$ is said to be a metric or a distance function on $X$ if $d$ satisfies the ...
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Intersection of interiors of sets in a partition of $\mathbb{R}^d$
Let $\mathcal{Q}=\{Q_1,...,Q_n\}$ and $\mathcal{P}=\{P_1,...,P_m\}$ be partitions of $\mathbb{R}^d$ with $n<m$. Assume all $Q_k\in\mathcal{Q}$ and $P_i\in\mathcal{P}$ are close and convex sets. ...
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Which metrics (on vector spaces) can be induced?
Is there a way to classify which metrics defined on vector spaces can be induced by a norm? ie. there exists norm $n: X\to \mathbb{R}$ on the vector space $X$ such that the metric $d(x,y)=n(x-y)$.
I ...
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Axiom of Choice in characterizing openness in subspace
Below is the typical characterization of open sets in a subspace $Y$ of a metric space $X$.
$E$ is $Y$-open iff there exists an $X$-open $S$ such that $E = S \cap Y$.
The forwards direction usually ...
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Does this increasing sequence of subsets of a bounded connected metric space $(X,d)$ terminates at some point at $X$?
This might be a silly question; this is where I'm stuck as to whether a metric-bounded set in a connected metric space is uniformity-bounded in the sense of Bourbaki. Let $(X,d)$ be a bounded ...
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