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.
16,332 questions
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What's the most logical decay profile *in a round space* of a force whose decay is approximated locally by the square of distance? [closed]
Question
What's the most logical decay profile in a round space of an attractive force whose decay is approximated locally by the square of distance ?
My attempt
Assume $S^3$ for now - so that's like ...
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Various types of Continuity in Metric space [closed]
Does anyone have a books/journal about continuity in metric space that i can carry for my thesis?
Especially about pointwise continuity, uniform continuity, holder continuity, lipschitz continuity, ...
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Difficulties with Set-Notation in Taxicab Geometry
The textbook I use introduced the Taxicab "circle" using the set-notation,
$$\{P\mid d_{T}(P,A)=1\}.$$
as the set of points, such that the distance between all the points and the center is ...
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Gromov-Hausdorff distance (defined by correspondence) is attainable for compact spaces?
In Dmitri Burago's book A course in Metric Geometry, he gives an equivalent definition of Gromov-Hausdorff distance using correspondences (Theorem 7.3.25)
For any metric spaces $X,Y$,
$$ d_{GH}(X,Y)=\...
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Bracketing number vs covering number
Let $(\mathcal{F}, \lVert\cdot\rVert)$ be a subset of a normed space of real functions $f \colon \mathcal{X} \to \mathbb{R}$ on some set $\mathcal{X}$. Give two functions $l(\cdot)$ and $u(\cdot)$, ...
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The isometry classes of compact metric spaces form a set? [duplicate]
I'm reading some stuff related to Gromov-Hausdorff distance. I have a question out of curiosity:
Does the isometry classes of compact metric spaces form a set $\mathcal{M}$ (but not a proper class)?
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Does every smooth manifold admit a generic Riemannian metric?
Today, I came across the notion of generic Riemannian metrics for the first time. Some googling around informed me of the "definition" of what it means for a Riemmanian metric to be generic (...
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$f+g,f-g,f\cdot g, \frac{f}{g}$ are continuous. ("Topology Second Edition" by James R. Munkres.)
I am reading Topology Second Edition by James R. Munkres.
In this section, we discuss the relation of the metric topology to the concepts we have previously introduced.
First, we want to show that the ...
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What is the optimal strategy to locate a hidden point within 1ft across a large area using only distance-change feedback rounded to the nearest 5 ft?
I've been trying to work through a math problem and wanted to see if it's feasible to solve within a certain number of moves.
A target point is placed at unknown coordinates ...
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Find the distance between these two sets.
Consider $\mathbb R$ with the Euclidean metric. Let
$$F=\mathbb N \quad \text{ and } \quad G:=\left\{n+\frac{1}{n} : n \in \mathbb N\setminus \{1\}\right\}.$$
These two sets are disjoint closed sets. ...
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Proof that categories of Hausdorff spaces and metric spaces are not cartesian closed
It is known that the category $\mathbf{Top}$ is not cartesian closed, see MSE/2969372. Specifically, the functor $\mathbb{Q} \times -$ does not preserve the coequalizers. Also, $\mathbb{Q} \times -$ ...
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Prob. 8, Sec. 30, in Munkres' TOPOLOGY, 2nd ed: Is this metric space first-countable? second-countable? separable? Lindelof? [closed]
Here is Prob. 8, Sec. 30, in the book Topology by James R. Munkres, 2nd edition:
Which of our four countability axioms does $\mathbb{R}^\omega$ in the uniform topology satisfy?
Here $\mathbb{R}^\...
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A Fixed Point Theorem in Complete Metric Spaces for Self-Maps Satisfying This Contractive Condition
Let $(X, d)$ be a complete metric space, and let $f \colon X \longrightarrow X$ be a self-map of $X$ for which there exists a real constant $h \geq 0$ such that
$$
d \big( f(x), f(y) \big) \leq h \max ...
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Is there a metric $d$ on $\mathbb{R}$ such that $\{d(x,y):x,y\in\mathbb{R}\}=[0,\infty)\cap\mathbb{Q}$?
There is a bijection $f:\mathbb{R}\to\mathbb{N}^{\mathbb{N}}$, so that we can define
$$d(x,y)=\begin{cases}
0, & x=y\\
2^{\min\{n\in\mathbb{N}:f(x)(n)\neq f(y)(n)\}} & x\neq y
\end{cases},$$ ...
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Is there a notion of distance between rooted trees based on minimal edit operations?
I am trying to formalise a structure where each system is represented by a rooted tree of possible future states. I would like to define a distance between two such trees based on the minimal “cost” ...
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