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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Parametric Equation of a unit circle when the angle between $x$-axis and $y$-axis is not $90$ degrees
I know in regular Cartesian coordinates the parametric equation for a unit circle is $x=\cos(\theta)$, $y=\sin(\theta)$, and if the $x$ coordinates are stretched by an amount $a$, and the $y$ ...
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Confusion about the criteria for a complete metric space to be a length space
According to corollary 2.4.17 from the textbook "A Course In Metric Geometry" we have
A complete metric space $(X, d)$ is a length space iff,
given a positive $\varepsilon$ and two points $x,...
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Can the twice-punctured plane be given a homogeneous metric?
A homogeneous metric on a space $X$ is one for which the isometry group acts transitively on its points (for all $x,y\in X$, there is an isometry $\phi$ of $X$ such that $\phi(x)=y$).
If we repeatedly ...
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If $f:X \rightarrow Y$ is an $\epsilon$-map between compact metric spaces then there exists a $\delta > 0$ such that $diameter(f^{-1}(Z)) < \epsilon$
First a map $f: X \rightarrow Y$ is called an $\epsilon$-map if it is continuous, onto and for any $y \in Y$, $diameter(f^{-1}(y)) < \epsilon$. Now I'm trying to find a $\delta > 0$ such that if ...
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Uniqueness condition for points that minimize the distance between two sets
Suppose that $\Omega \subset \mathbb{R}^N$ is a open bounded smooth domain and $(x_n) \subset \Omega$ is a sequence such that $d(x_n,\partial\Omega) \to 0$, as $n \to +\infty$, where $d(x_n,\partial\...
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Shift invariant metrics on the 1-dimensional torus
Let $\mathbb{T}:=\mathbb{R}/\mathbb{Z}$ be the 1-dimensional torus with the usual topology. It is well known that this is a metric space with the distance $d(x+\mathbb{Z},y+\mathbb{Z}):=\min_{n\in \...
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Inverse limit of arc-like spaces is arc-like
I came across this in Nadler's book, "Introduction to continuum theory".
First, for a collection of spaces $P$, a compact metric space $X$ is said to be $P$-like if for every $\epsilon > ...
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What is the completion with respect to a topology
I am reading the book "Vertex Algebras and Algebraic Curves: Second Edition" by Edward Frenkel and David Ben-Zvi.
In section 2.1.2 (page 26), two versions of Heisenberg Lie algebra are ...
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If A and C are nearer, is B nearer too?
Let $A, B, C$ be three colinear points in $\mathbb{R}^2$, and $P, Q$ any two points in $\mathbb{R}^2$ (on the same line or not). I am trying to prove (or disprove) the following lemma, where $d$ is ...
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What function minimizes the distance to its argument?
Let $d$ be a metric on $\mathbb{R}^n$, and $f: \mathbb{R}^n \to \mathbb{R}^n$ be a function.
I am interested in functions that satisfy the following property.
For all $x, y\in \mathbb{R}^n$,
$$
d(f(x)...
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Equivalence between Hausdorff metric and Area metric using openness criterion
Prove that the Hausdorff metric and the Area metric are not equivalent on the set of all bonded polygons, but are equivalent on the set of all convex polygons. The area metric is defined as the ...
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Is there distance (metric) between two points on the hyperreal line in Nonstandard analysis?
As is known, there is distance between two points on the real line. It's obvious.
But if we imagine hyperreal line (see pic.) then we'll have infinitesimals and their corresponding points.
For ...
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Upper semicontinuity of sequence of Hausdorff measures
This is exercise 12.2 of Measure theory and integration of Michael Taylor.
Let $(d_j)$ be a sequence of metrics on a compact space $X$ such that there exists $c,C>0$ with
$$ cd_0(x,y) \leq d_j(x,y) ...
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Triangle inequality property of a metric [duplicate]
A few years ago (around 8) I solved the following question:
Given the metric $d_*(x,y) = \text{min}(|x-y|, 1-|x-y|)$ in $X = [0,1)$, prove it holds the triangle inequality property.
I solved it as ...
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If there exists $A\subset X$ such that $X$ is equal to the closure of $A$ then $X$ is closed. [duplicate]
QUESTION:
Consider $\mathbb{R}$ with the standard topology and let $X\subseteq\mathbb{R}$. If there exists $A\subset X$ such that $\overline{A}=X$ then $X$ is closed?
MY ATTEMPT:
On one hand, $\...
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