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.
16,302 questions
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Relative compact set in metric space
Below is an equivalent way of stating relatively compactness ($\bar{A}$ is compact) in a complete metric space.
$\forall \epsilon>0$, there exists a compact set $K$ such that $\forall a \in A$, $d(...
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Let be X a metric space, A and B closed sets such that the union and intersection are arcwise connected, show that A and B are arcwise connected. [closed]
I have seen the same question, but talking about connected spaces. I would like to know how to do this with arc wise connected spaces.
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Pigeonhole principle based measure
Define the upper pigeonhole density of some $X\subset \mathbb R^d$ as
$\overline{pd}(X)=\limsup_{n\in \mathbb N} \inf_{S\subset\mathbb R^d, |S|=n} \sup_{x\in R^d} \frac{|S\cap (X+x)|}{|S|}.$
Can I ...
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Functions that map metrics to metrics [duplicate]
There is a classical example of a metric obtained from another metric:
$$\overset{\sim}{d}=\frac{d}{1+d}$$
where, over the same set $X$, the latter creates a bounded metric space that is complete iff ...
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How is $F_n$ defined inductively a closed set?
In Infinite Dimensional Analysis, in the proof of the result
Every closed subset of the Baire space $\mathcal N$ has a least element in the lexicographic order.
the authors considered a closed ...
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Subsequential limits of $(\cos(k))_{k\in \mathbb{N}}$
I am looking at a text book which is a reference in my educational background, and I usually find it to be a reliable source, but I am struggling with one of the proofs in it:
We'd like to prove that ...
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Continuity of piecewise application between Metric Spaces
I have the following exercise and I don't know if my proof is correct:
Let $(X,d_{X}),(Y,d_{Y})$ metric spaces with $X=X_{1}\cup X_{2}$
Let $f_{i}:X_{i}\to Y, \ i\in\{1,2\}$ continuous applications ...
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$f$ is continuous iff $f_{1},\dots,f_{n}$ continuous in Metric Spaces
I`m trying to do the following exercise for my general topology class:
Let $(X,d)$ a metric space and $B\subset\mathbb{R}^n$ with euclidean
metric. Let $f:X\to B$ and application determined by $f_{i}...
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Angle problems in two-dimensional CBB
Let $X$ be a 2-dim CBB($k$) space, and $AO,BO,CO$ are all geodesics, do WE have
$$\angle AOC=\angle AOB+\angle BOC\,?$$
I know that if $AC$ is geodesic, then $\angle AOB+\angle BOC=\pi$, but in ...
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How much can the Haar measure on an $n$-dimensional Lie group be expanded by a Lipschitz map?
I'll start with the question itself:
Let $G$ be an $n$-dimensional Lie group, $\mu$ be the Haar measure on $G$, $d$ be a translation-invariant metric on $G$ generating the topology on $G$. Let $f: G \...
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Levi-Civita connection for ${\mathrm{S}}^1$.
Let ${\mathrm{S}}^1$ be a circle with the radius $\frac{1}{2}$, which is a one-dimensional Riemannian manifold. I would like to give the chart on ${\mathrm{S}}^1$ by the affine $x$-real line and the ...
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Is $d(u,v):=\|u-v\|/\sqrt{\|u\|^2 + \|v\|^2 - \text{Re}\langle u,v \rangle}$ a metric on an inner product space?
Let $u,v$ be vectors of a inner product space $H$. I make the conjecture that the expression
$$
d(u,v) = \frac{\|u-v\|}{\sqrt{\|u\|^2 + \|v\|^2 - \text{Re} \langle u,v \rangle}}
$$
is a metric on $H$ (...
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Visualizing a basic triangle inequality in metric spaces
Let X be a metric space and A a subset of X.
I would like to visualize this basic inequality:
$$
\lvert d(x,A)-d(y,A) \rvert \le d(x,y)
$$
I tried drawing A as a blob, but I couldn't draw something ...
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Urysohn's Lemma with compact sets and uniformly continuous function
Problem: "By Urysohn's Lemma, we have that for any two disjoint closed sets A,B $\subset$ X in a normal topological space $(X,T)$, there is a continuous function $f: X\rightarrow [0,1]$ such that ...
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If $f:A\rightarrow \mathbb{Q}$ is a bijection and $d(a_i,a_j)=∣f(a_i)−f(a_j)∣=|q_i-q_j|$ , can $A$ have isolated points?
Let $A \subseteq \mathbb{Q}$ be a countable subset and let
$$
f : A \longrightarrow \mathbb{Q}
$$
be a bijection defined as $f(a_i)=q_i$.
Define a metric on $A$ by
$$d(a_i, a_j) = |f(a_i) - f(a_j)|.$$
...
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