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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Is there a bijection of $\mathbb{R}^n$ with itself such that the forward map is connected but the inverse is not?
Let $(X,\tau), (Y,\sigma)$ be two topological spaces. We say that a map $f: \mathcal{P}(X)\to \mathcal{P}(Y)$ between their power sets is connected if for every $S\subset X$ connected, $f(S)\subset Y$ ...
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Regarding metrizability of weak/weak* topology and separability of Banach spaces.
Let $X$ be a Banach space, $X^*$ its dual, $\mathcal{B}$ the unit ball of $X$ and $\mathcal{B}^*$ the unit ball of $X^*$. The following result is well-known
Theorem. If $X$ is separable, then $\...
user160185
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$f\colon I\rightarrow G$ and Gromov $\delta$-hyperbolicity
Please recall that $\left|\int_0^1 f(t)\,dt -w\right|\leq \int_0^1|f(t)-w|\,dt$. In general, let $(X,d)$ be a metric space. Given a function $f:I\to X$ let $m_f\in X$ be such that $d(m_f,w)\leq \int_0^...
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Can one cancel $\mathbb R$ in a bi-Lipschitz embedding?
Let $X$ be a metric space. Suppose that the product $X\times\mathbb R$ admits a bi-Lipschitz embedding into $\mathbb R^{n}$. Does it follow that $X$ admits a bi-Lipschitz embedding into $\mathbb R^{n-...
user31373
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What is the structure preserved by strong equivalence of metrics?
Let $d_1$ and $d_2$ be two metrics on the same set $X$. Then $d_1$ and $d_2$ are (topologically) equivalent if the identity maps $i:(M,d_1)\rightarrow(M,d_2)$ and $i^{-1}:(M,d_2)\rightarrow(M,d_1)$ ...
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Is there a metric on the plane for which (unit) circles are triangles?
If we use the Euclidean metric on $\mathbb R^2$, (unit) circles are circles; if we use the Manhattan distance or $\ell^1$ distance, (unit) circles are squares. We can also get polygons with an even ...
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Prove that every compact metric space $K$ has a countable base, and that $K$ is therefore separable.
Prove that every compact metric space $K$ has a countable base, and that $K$ is therefore separable.
How does the following look?
Proof:
For each $n \in \mathbb{N}$, make an open cover of $K$ by ...
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Is there any condition that makes a measure zero set necessarily countable?
Background :
Let us consider the Lebesgue measure space $(\Bbb{R}, \mathcal{L}(\Bbb{R}),m) $.
Here measurable set means Lebesgue measurable and measure means Lebesgue measure.
$\mathcal{S}\subset \...
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When is an Open Set Homeomorphic to the Interior of its Closure?
Let $X$ be a topological space and $U \subseteq X$ open. Then $U \subseteq \operatorname{int}(\operatorname{cl}(U))$.
I am looking for known assumptions on $X$ and $U$ such that one of the following ...
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Point set where each point has unity distance to all other points ($L_1$ metric)
I want to construct a point set where each point has the same (w.l.o.g., unit) distance to all other points in the $L_1$ metric. Example: The points $\left(\frac{1}{2},0\right)$, $\left(-\frac{1}{2},0\...
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Contracting subsets
Let $X$ be a (locally finite) metric graph (all of whose edges are length 1). A subset $A \subset X$ is contracting if there exists a constant $C \geq 0$ such that the nearest point projection on $A$ ...
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Non empty set with zero diameter
Let $A \subset X$ where $X$ is a metric space. by definition diam$(A) = \sup\{ d(x,y), x,y \in A\}$.
if $A$ is non empty and has zero diameter, can I conclude that $A$ is a singleton?
i reason as ...
user290300
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Is the barycenter of a convex curve in $\mathbb R^2$ Lipschitz with respect to the Hausdorff distance?
For a curve $C$, its barycenter is
$$\text{Bar}(C) = \frac{1}{\text{length}(C)}\int\limits_C x d \mathcal H^1(x).$$
Does there exist a constant $L$ such that for $C_1,C_2$ convex curves in the plane, $...
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Uniform Spaces: Completeness
Attention
This thread has been generalized to uniform spaces as general metric spaces.
Context
The context was the equivalence:
$$K\text{ compact}\iff K\text{ totally bounded, complete}$$
That is a ...
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What is the epistemological status of the usual proof(s) of Pythagoras' theorem?
Pythagoras' theorem has a variety of geometric proofs, such as:
I want to teach at least one of these proofs to my high school students, because it shows that the formula $\|(x,y)\| = \sqrt{x^2 + y^2}...
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