Metric spaces are sets on which you can measure the "distance" between any two points. The distance measurement is generally required to be symmetric (so distance from $A$ to $B$ is the same as distance from $B$ to $A$), positive for two distinct points, and obeying the triangle inequality.

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Prove that the distance metric between SETS satisfies the triangle inequality: $d(A,B)\leq {d}(A,C)+d(C,B)$ [on hold]

Prove that the distance metric between SETS satisfies the triangle inequality: $$d(A,B)\leq{d}(A,C)+d(C,B).$$ I can't seem to prove the fourth rule that this is a metric. Any help?
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3answers
94 views

Proving intersection of dense subsets of a metric space X is the isolated points of X.

Suppose X is a metric space. Let $\mathscr C$ denote the collection of all dense subsets of X. Show that $\bigcap\mathscr C $ = iso(X). Thus the question asks to prove that every dense subset of X ...
2
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2answers
20 views

Property of distance and adherence

Please how to prove that in a metric space $(E,d),$ for $A,B\subseteq E$ that $\forall x\in E, d(x,A)=d(x,\overline{A})$ and that $d(A,B)=d(A,\overline{B})=d(\overline{A},\overline{B})$ and ...
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Gelfand width of $l_1$ unit ball in $\infty$ metric

The $l_1$ balls in $l^N_p$ are well studied, but I cannot find anything about estimates of $d^m(B^N_1,l^N_\infty)$ except of $d^m(B^N_1,l^N_\infty)\geq C\min\{1,\frac{\ln(eN/m)}{m}\}$ which is not ...
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Equivalence of Theorems; the sphere on $\ell^2$ is finitely oscillation stable.

I just started reading the book Dynamics of Infinite-dimensional Groups, by Pestov, and right in the introduction the following theorem by Milman is cited: Let $\mathbb{S}^\infty$ denote the sphere in ...
3
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1answer
37 views

Gradient of the distance function

Let $\Omega$ be open, bounded subset of $\mathbb{R}^n$. Let $d(x):=dist(x,\partial\Omega)$ denotes the distance of the point $x\in\Omega$ from the boundary $\partial\Omega$. Define function ...
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1answer
54 views

Totally disconnected space in which some quasicomponents have interior?

Assume all spaces are metric. Question. Does there exist a space $X$ which is totally disconnected (the components of $X$ are singletons), yet some quasicomponent of $X$ has nonempty interior? I ...
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31 views

Elements with infinite roots in p-adic

Let $\mathbb{Q}_p$ the $p$-adic completion of $\mathbb{Q}$ and $$S=\{x\in\mathbb{Q}_p:1+x\mbox{ has n-th root in }\mathbb{Q}_p\mbox{ for infinite }n\in\mathbb{N}\}$$ I have to show that ...
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3answers
269 views

A not complete metric space?

Please how to prove that the space $\mathbb{R}$ endowed with the metric $d(x,y)=|e^x-e^y|$ is not a complete space? I don't find a Cauchy sequence but not convergent Please Thank you.
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1answer
53 views

show that $X$ is a complete metric space [on hold]

Let $\lambda$ be a real number and $X$ the set of all continuous real valued functions $\phi$ defined on $[0, \infty)$ which are such that the set $\{e^{-\lambda x}|\phi(x)|:x \ge 0\}$ is bounded. How ...
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49 views

How is a metric space a topological space? [duplicate]

I learned about metric spaces and topological spaces but I don't see how they correlate. How does a metric space follow the properties of a topological space.
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2answers
46 views

Prove that a set in a metric space cannot be both open and closed.

If I have a metric space $X$, and $E \subset X, E \ne X, E \ne\emptyset$. I want to prove that E cannot be both open and closed. I have two strategies, but I am not able to finish them: I assume ...
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1answer
45 views

Convergence of distances in metric space

If $(X,d)$ is a metric space, $(x_n)$ and $(y_n)$ are Cauchy sequences in $(X,d)$. How do i show that $(a_n):=d(x_n,y_n)$ converges? Here is what i did: Let $(x_n)$ and $(y_n)$ be Cauchy sequences, ...
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4answers
144 views

Absolute convergence in a metric space

Let $(X,d)$ be a metric space, $(a_n)$ and $(b_n)$ are sequences in $(X,d)$. If $\sum_{n=1}^\infty d(a_n,b_n)$ is absolutely convergent, what do I say about the convergence of $(a_n)$ and $(b_n)$?
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1answer
45 views

Metrics (Distances) on $\mathbb{F}$ Theorem Proof

I had a question regarding a Theorem I had come across that described metrics (distances) on ordered field $\mathbb{F}.$ Here it is: Theorem: If $\mathbb{F}$ is an ordered field, then $d(x,y)=|x-y|$ ...
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1answer
86 views

Is $d(x,y)=|x-y|^2$ a distance on $\mathbb{R}$?

Please how to prove that $d(x,y)=|x-y|^2$ is a distance on $\mathbb{R}$, I don't know how to solve the triangular inequality. Thank you.
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1answer
31 views

Linear bound on angles in an euclidean triangle.

I am trying to understand a proof in the book of Burago "A Course in metric geometry" (Lemma 10.8.13 page 383). I have difficulties with a certain inequality for the angle of euclidean triangles: ...
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3answers
56 views

Topology; Definition of the open ball and open sets confuses me

I just picked up T.W Gamelin’s book on topology. I started reading and got confused when I came to the definition of an open ball on the second page. It says $B(x;r) =$ All $y$ in the set $X$ such ...
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2answers
38 views

Continuous piecewise smooth curve

I cannot understand the definition of $\tilde d(p_1,p_2)$ here? Can anyone please explain it clearly?
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1answer
38 views

Subbases and half-planes

If $(X,d)$ is a metric space, it's easy to show that $H(x,y)=\{w\in X\mid d(x,w)>d(y,w)\}$ is open in the topology $\tau$ induced by $d$. Is, in general, $\{H(x,y)\mid (x,y)\in X\times X, x\neq ...
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60 views

At what conditions a compact metric space of covering dimension $n$ (on $\mathbb R^n$) is an n-manifold?

In the discussion to the MSE post an answer of @MatthewPancia with correction in a comment of @JasonDeVito would state: Every compact metric space of covering dimension $n$ can be embedded ...
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1answer
34 views

Euclidean metric formula

Is this the correct formula for the euclidean metric (in $R^4$ )? $g_E = dr^2 + r^2(d\theta^2 + d\phi^2 + d\tau^2 + \cos \theta d\tau d\phi).$ I have been doing some calculations that are wrong and ...
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Why is boundedness necessary in this proof that addition and scalar multiplication are continuous maps in the metric space induced by a norm?

Why is boundedness necessary in this proof that addition and scalar multiplication are continuous maps in the metric space induced by a norm? Suppose that $\lambda _n \to \lambda $, $\mu _n \to \mu ...
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How show the map $f:\mathbb R^2\rightarrow\mathbb R$, defined as $f(x,y)=x+y$ is continuous for all $(x,y)\in\mathbb R^2$?

Question: I want to show the map $f:\mathbb R^2\rightarrow\mathbb R$, defined as $f(x,y)=x+y$ is continuous for all $(x,y)\in\mathbb R^2$. Issue: I know how to prove this via the epsilon-delta way. I ...
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27 views

Show existence of a sub-sequence $(f_{n_k})$ which is uniformly convergent to a function in $C[0,1]$

Let $f_n:[0,1]\rightarrow R$ be a sequence of continuously differentiable function, Let $M>0$ be such that for any $0\le x \le 1$ and natural $n$, $|f_n(x)|$, $|f'_n(x)|<M$ Show existence of a ...
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1answer
36 views

A question about a perfect space and a linear order on it

Suppose I have a nonempty perfect Polish space $X$, and there's some linear order $<$ on it (it is not related to the topology on $X$ in any way). How can I prove that there is a point $y$ in $X$ ...
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1answer
49 views

Proving the existence of a sequence of polynomials convergent to a continuous function $f$.

I need to show that if $f$ is continuous function ($f:\mathbb{R}\rightarrow \mathbb{R}$), then there exists a sequence of polynomials which converges to $f$ on any compact subset of $\mathbb{R}$. I ...
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1answer
43 views

A geodesic metric space is a manifold on its own right. What are conditions for a Finsler space to be a manifold?

A geodesic metric space can locally be approximated by a vector space. This approximation provides it with a natural manifold structure. It means that geodesic metric space is more fundamental concept ...
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Product spaces $X = Y = \mathbb R$

Let $(X, d_X)$ and $(Y, d_Y)$ be metric spaces. It is defined for $d_{X \times Y} : X \times Y \rightarrow \mathbb R_+$ with $$d_{X \times Y}((x_1,y_1),(x_2,y_2)) := d_X(x_1, x_2) + d_Y(y_1, y_2)$$ ...
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1answer
47 views

Is this a metric on matrices?

In the set of $n$-by-$n$ reversible real matrices, decide whether $$d(A,B)=\ln (\lVert A^{-1}B\rVert\cdot\lVert B^{-1}A\rVert)$$ defines a metric and/or semi-metric. Can you please help me to solve ...
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1answer
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Example of equivalent but not strongly equivalent metrics

Please could someone show me an example of metrics $d$ and $d'$ that are not strongly equivalent but are equivalent? I read the Wikipedia article here but couldn't find an example. For completeness ...
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1answer
17 views

Continuity of evaluation maps in the topology of compact convergence on $C([0,\infty),\mathbb{R}^{n})$

I'm trying to prove that the evaluation maps $e_{x}:C([0,\infty),\mathbb{R}^{n})\rightarrow\mathbb{R}^{n}$ given by $e_{x}(f):=f(x)$ are Lipschitz-continuous with respect to the metric ...
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5answers
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concepts which is present in metric space but not in topological space

I want to know some concepts which is present in metric space but not in topological space. The one that first comes to mind is uniform continuity, equicontinuity i.e. concepts defined with some kind ...
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2answers
23 views

A problem similar to Banach fixed point theorem

a) Let $(X,d)$ be a complete metric space and let $T: X \to X$. Prove that if there exists a natural $n$ such that $T^n(x)$ (composition of $T$ $n$ times) is a contraction then $T(x)$ has a unique ...
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Prove ${x:d(x,p) < d(x,q)}$ is open in metric space $X$

$X$ is a metric space and $p \neq q$ $\in X$. I want to prove that $E=$ $\{x:d(x,p) < d(x,q) \}$ is open in metric space $X$. I think I can directly prove this by showing every point $x \in E$ ...
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1answer
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Metric Spaces: closure of a set is the set of all limits of sequences in that set

I am studying metric spaces and got confused about many different ways of defining the closure. Let $S$ be a subset of $M.$ Then, the closure of $S$ is $ \{x \in M : \forall \epsilon>0, \ \ ...
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Every metric space is a D-space.

I think it is correct, but I would like another pair of eyes to verify. Definition. An open neighborhood assignment is a function $f:X\to \tau$ such that $x\in f(x)$. Definition. A space is said to ...
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2answers
37 views

Continuity set of a difference of two upper semi-continuous real functions over a metric space [closed]

I wanted to know if we can get some properties of the continuity set of a difference of two upper semi-continuous real functions over a metric space? Or maybe for a restriction?.
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non-separable metric space and measurablility of its elements

I'm studying Skorokhod space, which consists of cadlag functions, and I encountered the following sentence: If a metric space $(\mathbb{S}, \mathcal{S}, d)$ is not separable, then functions that ...
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If $(X,d')$ is totally bounded and $d'$ and $d$ are topologically equivalent then $(X, d)$ is separable

I am trying to write something similar to the proof of If $(X,d)$ totally bounded then $(X,d)$ separable but I dont know how to use topological equivalence here. Any help?
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A metric space is complete if for some $\epsilon \gt 0$, every $\epsilon$-ball in $X$ has compact closure.

This is a problem from Munkres' Topology. Let $X$ be a metric space. (a) Suppose that for some $\epsilon \gt 0$, every $\epsilon$-ball in $X$ has compact closure. Show that $X$ is complete. (b) ...
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Why are every structures I study based on Real number?

I've been studying basic concepts of inner product vector space, normed vector space and metric space. And all the inner products, norms and metrics are defined to be real-valued functions in my ...
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Does nonexpansive mapping imply isometry in this case?

I have the following problem. I want to prove that there exists an isometric isomorphism: $$Lip_0(X) \equiv AE(X)^*$$ Here $(X, d)$ is a metric space, $Lip_0(X)$ is the space (a Banach space with the ...
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1answer
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Are pseudocompact metric spaces complete?

Is there a way to show that pseudocompactness on a metric space implies completeness directly (without using sequential compactness)?
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4answers
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Cauchy sequence in metric space

Give an example of a metric space such that a Cauchy sequence in $M$ that is not convergent. How can we give a example of that?
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1answer
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Isometries of Metric Spaces

For a metric space $(X,d)$, let $\def\Iso{\operatorname{Iso}}\Iso(X,d)$ denote the group of bijective isometries of $(X,d)$. Clearly, $\Iso(X,d)$ is a group under composition. Question: Let $X$ be a ...
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Nonobvious examples of metric spaces that do not work like $\mathbb{R}^n$

This week, I come to the end of the first year analysis, and suffer from a "crisis of motivation." With this question, I want to chase away my thought, "Why is it important to study the general ...
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If (X,d) is a separable metric space then there exists a metric d′ that is topologically equivalent to d and such that (X,d′) is totally bounded.

I know that this question Separability, total boundness and topological equivalence of metrics has been asked, but the only solution given is not valid. There is something I already knew: (Y, d2) ...
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A typical example of Homeomorphism

The set $\mathbb{R}^2-\{(0,0)\}$ with the usual topology is: (A) Homeomorphic to the open unit disc in $\mathbb{R}^2$ (B) the cylinder $\{(x,y,z)\in \mathbb{R}^3/ x^2+y^2=1 \}$ (C) the ...
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1answer
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A book with heuristics or general techniques used in real analysis?

I have been looking for a book with some good heuristics for real analysis and point set topology. Any ideas?