# 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.

10,158 questions
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### A uniformly continuous function with the supremum metric…

Question: Suppose that $C([0, 1])$ is the metric space of all continuous real-valued functions on $[0, 1]$, with the metric $d(f, g) := \sup_{x \in [0, 1]}|f(x) - g(x)|$. Let $f \in C([0, 1])$ such ...
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### Show that the interior of A in P is equal $(A\cup P^c)^o\cap P$

Let $A\subseteq P\subseteq X$, where $(X,d)$ is a metric space with distance function $d$. Define the interior of $A$ in $P$, written as $Int_P(A)$, to be the union of all sets $E\subseteq P$ such ...
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### detecting flares with persistent homology

Can persistent homology detect "flares" how does it do so, if it can. I know persistent homology can certainly find "loopy" structure, like noisy circles, but I'm not sure about "flares".
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### How to define a notion of Hausdorff homeomorphism?

A separable metric space is called fractal if its Hausdorff and topological dimensions are different. The Hausdorff dimension is not invariant by homeomorphism (see this post). Question: How to ...
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### metric space and limit point

Given $X$ metric space and $E$ is any subset of $X$. If $x$ is a limit point of $E$, for every $\epsilon>0$, prove that neighborhood of x contains infinitely many element I use the fact that ...
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### Connectedness of Metric Spaces [duplicate]

Let $(X, \rho)$ be a metric space which is compact suppose that for all $x \in X$ and $r>0$ $\overline{B_\rho(x,r)} =\{y \in X : \rho(x,y) \leq r\}$. Show that $B_{\rho}(x_0,r_0)$ is connected for ...
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### Union of connected sets with possible empty intersection

There is a result which states that if a collection $A$ of connected sets has a point $P$ belonging to every of those sets, then its union is connected I was wondering if this remains true if the ...
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### Is there a name for this notion of “radius of compactness” in a metric space?

I was proving some result about Riemannian manifolds that led me to introduce the following definition: Let $M$ be a metric space and $x \in M$. Define the "radius of compactness" $RC(x)$ to be the ...
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### Example of noncompact space in which every real valued continuous function on it is uniformly continuous

I wanted to find Example of non-compact metric space $(X,d)$ such that every real-valued continuous function is uniformly continuous My attempt: $X$ is an infinite set $d$ is a discrete metric. ...
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### Distance in a Riemannian submanifold (compact)

Let $M,S$ be compact connected Riemannian manifolds such that $S\subset M$ (injectively immersed). Denote $d_M$ and $d_S$ their respective Riemannian distances. Is $d_M$ restricted to $S\times S$ ...
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### Sequence of points in a nested sequence of sets converges to the point in the limit of the sequence of sets.

Let $B_n$ be a decreasing sequence of compact subsets of a metric space convergent to compact set $B$. That is $B_{n+1}\subseteq B_n$ for all $n$ and $\bigcap\limits_{n\geq 1}B_n = B$. Let $b_n\in B_n$...
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### How can I prove that my function d(x, y) is a proper metric?

I have points that are characterized by a timestamp and a location, so 3 dimensional points, one temporal x, and two for the location coordinates. My function $d(x, y)$ is defined as follows: d(x, ...
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### Least possible distance distortion for a map

The background to the question is that I would like to figure out how much a map of Europe must distort distances. Let us try to formulate this mathematically. Say I have a closed subset, $D$, of ...
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### Gradient and Directional Derivative in Riemannian Manifold

Probably this question is too dumb to be asked, but I am an engineer trying to learn differential geometry, please go easy on me. I am trying to understand that, in Riemannian space, gradient ...