# 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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### Updated Gorelik principle

One version of the Gorelik principle is the following: Let $E,X$ be Banach spaces and suppose $U:E\to X$ is a Lipschitz isomorphism (that is, a Lipschitz bijection whose inverse is also Lipschitz). ...
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### Let $X_{n}$ and $Y_{n}$ are two bounded sequence. I need to prove $\sup_{n\in\mathbb N}|X_{n}-Y_{n}|=0$ iff $X_{n}=Y_{n}$ for all $n$

Let $X_{n}$ and $Y_{n}$ are two bounded sequence. I need to prove $\sup_{n\in\mathbb N}|X_{n}-Y_{n}|=0$ iff $X_{n}=Y_{n}$ for all $n$. I want this proof. Because if you see this in metric space then ...
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### Directly proving a metric function is continuous

Let $(X,d)$ be a metric space an $p\in X$ be any point. I want to show that a function $f:X\to\mathbb R$ defined by $f(x)=d(x,p)$ is continuous. My attempt: Let $G\subseteq\mathbb R$ be open and ...
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### What is wrong with my answer: In metric space M, If A is dense in S and B is open in S, prove $B \subseteq \overline{A \cap B}$

This was a test question. My answer: B is open in S, so $$B \subseteq \overline{B} \subseteq S \text{(this is what was marked incorrect) }$$ A is dense in S, so $$A \subseteq S \subseteq \overline{A}$$...
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### Is there such a thing as a continuous metric space? [closed]

If one looks at the set of real numbers equipped with absolute value metric, then we notice that any open interval of any size is non empty. I was wondering if continuity of any metric space could be ...
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### Exercise 8, Section 3.2 - Metric Spaces: A Companion to Analysis

The question I am trying to solve is the following from Metric Spaces: A Companion to Analysis: Let $X$ be a complete metric space. Let $A$ be a nonempty set in $X$. Show that $A$ is homeomorphic to ...
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### Injective path in path-connected space [duplicate]

In any path-connected metric space, can we connect any two points with an injective path? This seems possible, for instance by "removing" the points where the loop is not injective, but I ...
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### For $I:(X,d_1 )\to(X,d_2)$, $I(x)=x$, proving $I$ is continuous iff $\tau_{d_1}=\tau_{d_2}$

Problem.src) Let $d_1$ and $d_2$ two metrics. Let $I : (X, d_1) \to (X, d_2)$ be given by $I(x) = x$. Then is the following true? $$\text{I is continuous} \quad\iff\quad \tau_{d_1} = \tau_{d_2}$$ ...
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### Every probability measure on a metric space is regular

In Billingsley's book Theorem 1 says the following: Let $S$ be a metric space equipped with the metric $\rho$ and be $\mathcal{S}$ be the corresponding Borel $\sigma$-field. Let $\mu$ be a probability ...
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I am studying Analysis by my own and I have a problem in the following question. Suppose $X=\mathbb{R}$ and define the function $d: \mathbb{R} \times \mathbb{R} \rightarrow [0, +\infty)$ as follows: $$... 0 votes 1 answer 93 views ### Proof that pointwise continuity implies uniform continuity for functions on compact sets Theorem: Let X be a compact metric space and f : X→\mathbb{R} be such that f is pointwise continuous for every x \in X. Then f is uniformly continuous on X. I was trying to prove this ... -1 votes 0 answers 83 views ### Homeomorphism between a plane with a taxi-cab metric and railways metric? Is a plane with a French railways metric homeomorphic to a plane with a taxi-cab metric? Here are French railways and taxicab definitions As far as I know, to show homeomorphism, one needs to identify ... 0 votes 1 answer 23 views ### Series bounded by uniform convergent series is uniform convergent? I was reading about the Weierstrass M-test, and I was wondering about the following: Suppose that (f_{n}) is a sequence of real-valued functions defined on a set A, and there exists (g_{n})_{n ... 2 votes 1 answer 19 views ### Prove that in a normalized space the function reaches its infimum. Let X be a normalized space, x, y \in X. Function \varphi:\mathbb{R}→\mathbb{R} is given by the formula \varphi(t)=||x−ty||. Prove that it reaches its infimum. \textbf{My idea:} Consider the ... 1 vote 0 answers 39 views ### Riemannian metric on fixed rank manifold I know that one can define metrics on the manifold of SPD matrices$$ \mathcal{S}^n = \{ A \in \mathbb{R}^{n\times n} \ | \ \text{A positive semi-definite} \} $$like the Log-Euclidean metric or the ... 1 vote 0 answers 60 views ### Limiting proportions of a matrix "metric" This question is based on an earlier posting on the Wikipedia math reference desk, although in that posting the "equals" proportion and the "greater than" proportion were ... -2 votes 1 answer 63 views ### Nature of metric spaces Let X be a metric space, and Y be a closed subset of X such that the distance between any two points in Y is at most 1: is Y compact ? is any continuous function Y\to \mathbb{R} bounded?... 1 vote 0 answers 26 views ### If x1,...,xn are points of a metric s., must there exist a normed s. (E, ||.||) and y_1,...,y_n\in E:d(x_i,x_j )=||y_i-y_j|| with all i,j=1,...,n? [duplicate] If x1, x2,...,xn are points of some metric space, does there necessarily exist such a normed space (E, ||.||) and y_1,y_2,...,y_n\in E that d(x_i,x_j )=||y_i-y_j|| with all i,j=1,2,...n? I am ... 2 votes 1 answer 53 views ### If \lim_{n\to\infty}A_n=A\neq\varnothing as set-theoretic limit, is it true that \lim_{n\to\infty}\mathrm{diam}(A_n)=\mathrm{diam}(A)? Let (X, d) be a metric space with topology induced by the metric d, and (A_n)_{n=1}^\infty be a sequence of sets in X such that \lim_{n\to\infty}A_n = A \neq \varnothing as a set-theoretic ... 2 votes 1 answer 54 views ### A Lipschitz map universal with respect to factorization through it of Lipschitz functions. Let (X,d) be a metric space. Given a metric space (Y,\rho) and a 1-Lipschitz map \varphi\colon X\to Y, we say that a 1-Lipschitz function f\colon X\to \mathbb{R} factors through \varphi ... 1 vote 1 answer 54 views ### Exercise 2 Chapter 4.1 - Magnus I am trying to solve Exercise 2 (Section 4.1.6, Page 129) from Robert Magnus "Metric Spaces: A Companion to Analysis". I have tried to prove item (a), but I am a unsure on how to approach ... 2 votes 0 answers 73 views ### How to prove that  \left \{ \prod_{i=1}^{n} U_i: U_i \text{ are open in } X_i \right \}  is a base for the product topology? I'm trying to prove that$$ B = \left\{ \prod_{i=1}^{n} U_i : U_i \text{ are open in } X_i \right\}  is a basis of the product topology. I was trying to use the subbase of the product topology, that ...
I have become interested in metrizability theorems and their generalizations to other notions of a metric. Consider an ordered pair $(M,d)$ and a total order $(T,\leq)$ with least element $0$ where $M$...
From Baby Rudin: Let $E$ be a non-empty subset of metric space $X$ and let $S = \{ d(x,y) | x,y \in E \}$ Then the diamater of $E$ is the least upper bound of $S$. Question: Don't we need that the ...