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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.

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Positivity and Symmetry from defination of Metric

I can across definition of metric as X is set $d:X\times X\to \mathbb R$ satisfying 1) $d(x,y)=0 $ iff $x=y$ 2) $d(x,y)\leq d(x,z)+d(y,z) , \forall x,y,z\in X$ I wanted to prove positivity and ...
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How do I prove that a closed subset of a compact set is compact directly from the definition? [duplicate]

I'm trying to prove that a closed subset of a compact set is compact directly from the definition but I'm not sure how to proceed. This is what I have so far: Let $K$ be a compact set and let $A$ ...
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Every isometry is Lipchitz-continuous

Let $(X, d_X)$ and $(Y, d_Y)$ be metric spaces, $f:(X,d_X) \rightarrow (Y,d_Y)$ be an isometry. Then $f$ is Lipchitz-continuous. Attempt: Suppose that $f$ is an isometry. Then for all $x_1,x_2$ in $...
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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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$d(x,y) = |f(x) - f(y)|$ on $\mathbb{R}$

Given any map $f: \mathbb{R} \to \mathbb{R}$, define the following function $$d(x,y) = |f(x) - f(y)|$$ for $x,y \in \mathbb{R}$ It seems to me - please confirm - that as soon as $f$ is injective, $...
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Rigorous construction of partition induced by minimal covering transformation

This is closely related to a previous question I recently asked. This was about the definition of a locally lifted measure described by Keane in his seminal paper on $g$-measures [1]. I received a ...
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Prove that $f: X \rightarrow R$ is continuous with respect to the metric $d_1$ on $X$ iff it is continuous with respect to the metric $d_2$ on $X$.

Let $X$ be a set and $d_1, d_2$ be metrics on $X$ so that for constants $m,M > 0$ and any $x,y \in X$ we have $md_1(x,y) \leq d_2(x,y) \leq Md_1(x,y)$ Prove that $f: X \rightarrow \mathbb{R}$ is ...
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Every surjective isometry on a Hilbert space is indeed a unitary operator

I have a little bit confused on unitary operators and surjective isometries on a Hilbert space. I think it is quite clear that A operator is unitary if and only if it is a surjective isometry. ...
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Closedness of Bi-Lipschitz embedding's image

Suppose that $(X,d)$ and $(Y,\rho)$ are complete and separable metric spaces. If $\phi:X\rightarrow Y$ is a bi-Lipschitz embedding, is it the case that $\phi(X)$ is closed in $(Y,\rho)$?
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Is the metric space of Lipschitz Function with $d_{\infty}$ complete?

This is my real analysis homework. Define $$\mathcal{L} = \{f : [a,b] \to \mathbb{R} : f \text{ is a Lipschitz function }\}$$ and the metric $$d_{\infty}(f,g) = \sup\{|f(x)-g(x)| : x \in [a,b]\}$$ ...
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Let $f:K \rightarrow N$ be a continuous function from a compact $K$. Show that $f$ is uniformly continuous

I'm having trouble finishing this. One approach that I made is this: Let $\epsilon > 0$. Then, since $f$ is continuous, for every $x \in K$ exists $\delta_x > 0$ such that $d(x, x')<\delta_x ...
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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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Open and closed sets and continuous functions

Suppose that $(X,ρ)$ is a metric space, $f:(X,ρ)\rightarrow R$ a continuous function and $D$ a dense subset of $X$ so as $f(D)$ finite. Prove that: (i) The range $f(X)$ is finite (ii) For every $t\...
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Cantor's Intersection Theorem iff complete metric spaces?

If a metric space is complete, we know that the Cantor's Intersection Theorem holds. Does the converse also hold? And if not, what is a suitable counterexample for the same? Also, if the converse ...
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Contractable Metric Spaces Homeomorphic to Euclidean Space

Is there a characterization of all metric spaces which are homeomorphic to a contractable subset of Euclidean space? This question is cross-referenced here.
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Is $\inf A $ and $\sup A $ belong $\bar A$?

Let $A$ be a anonempty and bounded subset of $\mathbb{R}$. Now take $A= (0,1)$ in the discrete topology of $\mathbb{R} $. My question is that : Is $\inf A $ and $\sup A $ belong $\bar A$ ? ...
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Arc connectedness implies connected

I've seen that are one or two questions like this, but I'm not fully convinced that are right. Both pretty much say the same thing, so I leave the easier to understand: path connectedness implies ...
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Is every metric space compact? [duplicate]

I am referring to Rudin's definition 2.32 of compactness here: A subset K of a metric space X is said to be compact if every open cover of K contains a finite subcover. Obviously X is a subset of X ...
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“Generalized” proof of the limit of a sum

Goal: to try to unify the sum of limits for $\mathbb{R}$, $\mathbb{C}$, and $\mathbb{R}^n$ with $n\ge 2$. Let be $(X,d_X)$ and $(Y,d_Y)$ two metric spaces and $f,g:D\to Y$, with $D\subseteq X$ two ...
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Metrics on $\mathbb{N}$ such that $(\mathbb{N}, d)$ and $(\mathbb{N}, d')$ are not homeomorphic

I want to find two metrics $d$ and $d'$ on $\mathbb{N}$ such that $(\mathbb{N}, d)$ and $(\mathbb{N}, d')$ are not homeomorphic, but I'm having trouble doing so. I've tried using the discrete metric ...
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Cauchy sequence with $\{ x_n : n \in \mathbb{N} \}$ not closed converges

Suppose $(x_n)_{n \in \mathbb{N}}$ is a Cauchy sequence and $A = \{ x_n : n \in \mathbb{N} \}$ not closed. Show that there exists $x \in X$ such that $x_n \longrightarrow x$. Since $A$ is not closed:...
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Metrizable in infinite number of ways

The question states that if a topological space is metrizable it is metrizable in infinite number of ways. Of course scaling the distances by any positive number will do the trick. But i want to ...
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Distance from a point to a convex set

Consider a sphere, which is obviously a convex set. Consider any point outside the sphere, and say I want to find the minimum distance to this set. In this case I can intuitively see that the closest ...
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$f \in \mathcal{C}(X, X)$ on metric space with $\sum_{n=1}^\infty d(f^n(x), f^n(y)) < \infty$ has a fixpoint

Let $X$ be a complete metric space and $f : X \to X$ continuous such that $$\sum_{n=1}^\infty d(f^n(x), f^n(y)) < \infty $$ for all $x, y \in X$, where $f^n$ means $f \circ \ldots \circ f$ $n$-...
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Are singletons closed or open?

"Exercise 1. Show that if $X$ is equipped with the discrete metric $d$ then every subset of $X$ is both open and closed. Deduce that any function $f : (X, d) → (Y, dY )$ is continuous." My lecturer ...
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Boundary of Set defined with $\sin(\frac{1}{x})$

Let $S=\{(x,y) \in (0,\infty) \times (-1,\infty) \mid y \geq \sin(\frac{1}{x})\}$ in $(\mathbb{R}^2,\lVert \cdot \rVert_\infty)$. I think that the boundary $\partial S= \{ (0,y) \mid y \in (-1, \infty)...
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Prove that $d(a,b) \le \inf ~\{ ~d(a,s)~|~s \in S \} + \inf ~\{ ~d(b,s)~|~s \in S \} + \sup ~\{~d(b,s)~|~s \in S\}$. Error in proof?

Suppose $(X,d)$ is a metric space and $a,b \in X, S \subseteq X, S \ne \{\phi\}$. Then, prove that $d(a,b) \le \inf ~\{ ~d(a,s)~|~s \in S \} + \inf ~\{ ~d(b,s)~|~s \in S \} + \sup ~\{~d(s_1,s_2)~|~...
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Two disjoint closed sets $A,B$ with $B$ compact, show $d(A,B) > 0$ Verify my proof

Two disjoint closed sets $E,F$ with $E$ compact, show $d(E,F) > 0$. So, with compactness we get a few things, that every sequence has a convergent subsequence and we can use the extreme value ...
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Complete quotient metric space

Let $(X,d)$ a metric space and $\sim$ a equivalence relation such that : $\forall x\in X$ : $[x]=\{y\in X \vert y \sim x \}$ is closed. If $[x] \neq [y]$ : $d([x],[y])=d(a,[y]), \forall a\in[x]$ ...
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Proof that the inverse image of a single element is a discrete space

Let $f: X \rightarrow Y$ be a local homeomorphism. I want to prove that, for each $y \in Y$, the fiber $f^{-1}(y)$ is a discrete set, or discrete space (Is there any difference between these two last ...
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Why is $x_0+\frac{r_0}{2}x \in B(x_0,r_0)$, $\|x\| \leq 1$?

Why is $x_0+\frac{r_0}{2}x \in B(x_0,r_0)$? Where $B(x_0,r_0) \subset E$ and $x \in E$, $\|x\| \leq 1$. $E$ is Banach (except that this probably doesn't matter here).
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Prove that this space is compact/not compact by using Arzelà–Ascoli theorem

I need to tell if this space/set is compact in $C[0,1]$ $x_n(t) = t^n, n ∈ N$ Following Arzelà–Ascoli theorem, the set is compact when it has Uniform boundedness and Equicontinuity, is it correct? ...
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Prove that metric space is complete/incomplete

I have a function $\rho(x,y) = |x^{1/3} - y^{1/3}|$ and I need to prove if the function is metric, and if it is, the next step is to prove if this metric space is complete. So metric is a function ...
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1answer
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Product topology. Clarification needed

Please consider the statement and it's proof below: The author further says that : Members of the product topology can all be expressed as union of products, but most members of the product topology ...
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1answer
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Continuity of Identity Functions Between Metric Spaces

I'm studying through Mendelson's Introduction to Topology and have a question on proving continuity. First, let's define two distance functions. Let $x= (x_{1}, ..., x_{y})$ and $y=(y_{1}, ... , y_{n}...
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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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Maximal Domain of Continuity

Let $f:(X,T)\rightarrow (Y,S)$ be a function between Polish spaces. Does there necessarily exists a (non-trivial: ie not a point or the emptyset) open subset $\tilde{X}\subseteq X$, on which $f|_{\...
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Metric Space defined by an Infinite Sequence of Metric Spaces in this case not a Metric Space

I just started my first book on Topology, Introduction to Topology by Bert Mendelson (Second Edition), and the second chapter is about Metric Spaces. The last problem of Section 2, Problem 2.9, is as ...
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Connected metric space of first category

Question: Does there exist a connected metric space of first category (i.e. it can be written as a countable union of nowhere dense subsets)? No such example is given in the book Counterexamples ...
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On the complete metric space

Let $\mathbb N$ be the set of all natural numbers. Set $$d(m, n) = \left\{\begin{array}{ll} 0, &\text{if } m = n, \\ 1+ \dfrac{1}{m+n}, &\text{if }m\neq n. \end{array}\right.$$ Prove that $(\...
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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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1answer
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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 ...