Metric spaces are sets on which a metric is defined. A metric is a generalisation of the concept of "distance". Metric spaces should not be confused with topological spaces.

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For a nonempty subset $A$ of a metric space, $x \in \overline{A}$ iff $d(x,A) = 0$.

Let $(X, \rho)$ be a metric space and $x \in X, A \subset X$ ($A \neq \varnothing$). Then $x \in \overline{A}$ iff $d(x,A)=0$. I am facing difficulties showing that $d(x,A)=0$ implies that $x ...
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Does a connected countable metric space exist?

I'm wondering if a connected countable metric space exists. My intuition is telling me no. For a space to be connected it must not be the union of 2 or more open disjoint sets. For a set ...
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contraction mapping unique solution

I am currently revising metric spaces and have come across a question slightly different to ones i am used to and would appreciate any hints as to how to attempt this question here is a photo of the ...
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Why don't clopen sets pose problems in the “preimage” definition of continuity, and in the definition of a topology?

Recently I have been wrestling with the reasoning behind why clopen sets do not lead to contradictions when we define continuity in terms of open/closed sets, the topology of metric spaces, and other ...
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Given $2$ closed subsets whose union and intersection are path connected. Show that each subset is path connected.

Let $A$ and $B$ be $2$ closed subsets of a metric space $E$. Suppose $A \cap B$ and $A \cup B$ are path connected. Show that $A$ and $B$ are path connected. Proof: it suffices to show that A is path ...
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complete spaces and the baire principle

There is this problem that I can't get an answer to. Begin with $(X,p)$ a complete metric space, with no isolated points. (This means that every point is an accumulation point, right?) Now prove ...
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Baire principle with open and dense subsets-edited

This is a question on the Baire principle for metric spaces. Let $X$ be a COMPLETE metric space without isolated points. Prove or disprove that, every sequence $(O_n)$ of open and dense subsets of X ...
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Is there a minus thickening operator on a metric space?

Let $S$ be a metric space and $A$ a subset. For some $\varepsilon>0$ define the $\varepsilon$-thickening of $A$ as $$A^{\varepsilon} = \left\{p \in S \mid \exists q \in A \;\;\text{with}\;\; ...
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Exercise from *Apostol's Mathematical Analysis*:

Exercise from Apostol's Mathematical Analysis: A point $x\in \Bbb R^n$ is a condensation point if every ball has the property that $B(x)\cap S $ is uncountable. Assume that $S\subset \Bbb R^n$ and ...
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Subspace or not?

I'm studying Functional Analysis and I got a doubt about the following theorem: Let $H$ be a prehilbert space and $S \subset H$ complete and convex. Then, $\forall x \in H$ there exists a unique ...
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Prove wrong the following statement about metric spaces and completeness

Statement: Given the condition: $d(x,y)^2 \leq g(x,y) \leq d(x,y) \space \forall \space x,y \in M$ If $(M,d)$ is complete then $(M,g)$ is complete Question: Prove or provide a counterexample to ...
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Isosceles Triangles in Hilbert Spaces and Metric Spaces generally

In what types of metric spaces $\langle X, d \rangle$ is it possible to do the following? Task: For any two points $x, y \in X$ such that $d(x,y) \leq 2\epsilon$, find a third point $z$ such that ...
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Determining compactness and completeness of metric space

Metric Space: (M,d) Set: $M = \{ (x,y)\in \mathbb{R}^2:y>0 $ or $ x=0=y \}$ Metric: $d((x,y),(a,b)) = $min$\{ $max$ \{ |x-a|,|y-b| \},y+b \}$ $\space$ completeness: $\lim_{n ...
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Can there be metrics on sets of random variables?

First off - I do not know much probability theory, so please pardon me if this question is nonsensical. The question arose from the following thought: can I make the expectation function continuous, ...
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Properties of oscillation

Let $(X,\tau)$ be a topological space, $(M,\operatorname{d})$ a metric space and $f:X \to M$. Then Definition: The oscillation of $f$ at $x \in X$ is $$ \omega_f(x) = \inf_{x \in U \in ...
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Difference between completeness and compactness

According to Wikipedia: A metric space $M$ is said to be complete if every Cauchy sequence converges in $M$ $ $ A metric space $M$ is compact if every sequence in $M$ has a subsequence ...
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Definition of compactness of metric spaces

In my lecturer's notes it says the following: Let $A \subseteq M$ and let $B = \{U_i : i \in I\}$ be an open cover of $A$. When determining the compactness or not of $A$, we might question whether it ...
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is finite product of complete metric spaces is the same as completion of finite product?

The following question seems trivial, but I want to make sure I do not have a mistake. Suppose I have $X,Y$ metric spaces. Denote $\overline{X\times Y}$ the completion of the product. If ...
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Intersection of closed balls [duplicate]

How does one find a decreasing sequence of closed balls ( not necessarily concentric) in a complete metric space whose intersection is empty?
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Is the function $d(x,y) = \frac{\|x-y\|}{\|x\|\|y\|}$ a metric?

$d$ is defined for all $x,y \in \mathbb{R}^2 - \{0\}$. It's clear that $d(x,y) = 0 \iff x=y$ and $d(x,y)=d(y,x)$ I am having issues with triangle inequality. I couldn't find a counterexample for ...
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Metric in the projective space $P^n$

Let $S^n = \{x \in \mathbb{R}^{n+1}; \langle\, x,x\rangle = 1 \}$. $P^n$ is the set of all unordered pairs $[x] = \{x,-x\}$, $x \in S^n$. I'd like to prove that $d([x],[y]) = \min ...
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Examples about open, closed and neither open nor closed subsets?

$B[0,1]=\{f| f:[0,1]\to\mathbb{R},\ f \text{ is continuous}\}$, $f,g \in B[0,1]$ $d(f,g)=\sup\{|f(x)-g(x)|, x \in [0,1]\}$ I need 6 examples (2 for each) about open subsets, closed subsets and ...
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maximum distance in between points in taxicab metrics

Let's define distance between two points $(x_1, y_1)$ and $(x_2, y_2)$ as $$|x_2-x_1| + |y_2-y_1|$$. There are given some points. I think how to find maximum distance between two arbitrary points ...
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Open compactification of metric space

Suppose I have a separable metric space $X$. I wanted to ask if there exists a separable metric compactification of this space $\overline{X}$, s.t. $X$ is considered open in $\overline{X}$.
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basis of neighborhoods of zero in Schwartz Space

I have the following question: In $S(\mathbb{R}^n)$, and for $\alpha, \beta\in \mathbb{N}^n $, we defined \begin{align} ρ_{\alpha,\beta}(\phi) = ...
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Prove for $x \in \mathbb{X}: x \in \bar{A} \leftrightarrow d_A(x)=0$

For $A \subset \mathbb{X}$ non empty and $x \in \mathbb{X}$ define the distance of x to A by $$d_A(x)=inf_{a \in A} d(x,a)$$ I am trying to prove for $$x \in \mathbb{X}: x \in \bar{A} \leftrightarrow ...
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Show for $x,y \in \mathbb{X} : |d_A (x) - d_A (y)| \leq d(x,y)$

For $A \subset \mathbb{X}$ non empty and $x \in \mathbb{X}$ define the distance of $x$ to $A$ by $$d_A (x) = \inf \limits _{a \in A} d(x,a)$$ I am trying to show for $$x,y \in \mathbb{X} : |d_A (x) ...
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Prove that if $f$ is continuous on a compact set then it is uniformly continuous

Prove that if $f$ is continuous on a compact set then it is uniformly continuous. Proof: Let $f:A\rightarrow \mathbb{R}$ be a continuous function and let $A$ be a compact subset in a metric ...
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Density In The Theorem/Proof of The Stone-Weierstrass Theorem

Yet again, I have a question that I could use some help with. Note that almost everything can be found in C. Pugh's, Real Mathematical Analysis (soft-cover, 2nd Edition, ISBN: 978-1-4419-2941-9); ...
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Let $U\subset \mathbb R^n$ and $Q=\{(x_1,…,x_n)\in U\mid x_1=1\}$. Show that $Q$ is not open and $\partial Q=\emptyset$.

Let $U\subset \mathbb R^n$ and $Q=\{(x_1,...,x_n)\in U\mid x_1=1\}$. Show that $Q$ is not open and $\partial Q=\emptyset$. To me it looks obvious, but how can I write a proof properly ? This is my ...
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An open ball in $C[0, + \infty)$

Consider the space $C[0, +\infty)$ of all continuous, real-valued functions on $[0, + \infty)$ with metric $$ d( \omega_1, \omega_2 ) = \sum_{n=1}^{\infty} \frac{1}{2^n} \max_{t \in [0,n]} ( \min ...
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Infinite metric space has open set $U$ which is infinite and its complement is infinite

Let $(X,d)$ be a metric space where $X$ is an infinite set. Prove that the space has an open set $U$ such that both $U$ and its complement are infinite sets. I have considered if $d$ is the ...
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A symmetric function

While working on a research problem on fuzzy metric spaces, I came across a special symmetric function $F_n:X^n\times (0,\infty)\to [0,1]$ i.e. \begin{equation*} ...
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In a complete metric space with no isolated points , show that the intersection of open and dense sets with a countable set is non-empty.

Let $(X,d)$ be a complete metric space with no isolated points. I want to prove that for every $(Gn)$ sequence of open and dense subsets of $X$ and for every countable set $A$$\subseteq$$X$ we have ...
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Sum of Closed Subsets of $\mathbb{R}$ [duplicate]

If $A,B\subset \mathbb{R}$ are closed in $\mathbb{R}$, is $A+B$ also closed in $\mathbb{R}$? I think it is not, but could not find a counter example: any suggestions?
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Hausdorff dimension calculation of union of sets

$F$ is a Cantor set in $(0,1)$, $\dim_HF=1/5$. What's the $\dim_HE$ where $E=(F×R)\cup(R×F)$? By the product properties, I know that and $\dim_H(F×[0,1])=6/5=1+1/5$, which is the sum of hausdorff ...
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Is squared Euclidean distance a metric? [duplicate]

Is squared Euclidean distance a metric? In particular does it obey triangle inequality? I think no, but cannot find a counterexample. Edit: Does this (not obeying the triangle rule) happen only when ...
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1answer
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The complement of a first category set in X is a set of second category.

Let X be a complete metric space. Then the complement of a first category set in X is a set of second category in X. What is explain in my class is "if the complement of a first category set is a set ...
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Suppose $f : X \to Y$ is a (continuous) bounded map.Does this implies that $f$ is uniformly continuous?

It's well known that if $ f : \bf (X,d) \to \bf (Y,e) $ is a uniformly continuous function then $f$ maps bounded set to bounded set.Does the converse hold ? More Precisely, Suppose $f : X \to Y$ ...
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Prove that two closed and disjoint sets, have open disjoint super-sets with dist metric .

For starters let's define the distance between a point $x$ and a set $A\subseteq X$ of a metric space $(X,d)$ as follows: $$\text{dist}(x,A)=\inf\{d(x,a):a \in A\}.$$ Now let's assume that $A,B$ are ...
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Why the word 'space' is used with metric space?

I am new to math. According to the Wikipedia, A metric space is a set for which distances between all members of the set are defined. I have a silly question: Why they used the word 'space'? Why not ...
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$S(f) = \underset{x \in X}{\sup } f(x)\ $ is continuous.

I'd like to prove that the function $ S: \mathcal{B} (M;\mathbb{R}) \rightarrow \mathbb{R}$, $S(f) = \underset{x \in X}{\sup } f(x)\ $ is continuous. $\mathcal{B}(M;\mathbb{R})$ is the set of all ...
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Where can I learn more about “approximate isometries”?

Let $Y$ and $X$ denote metric spaces and $f : Y \leftarrow X$ denote a function. Definition 0. Call $f$ an approximate isometry iff for all $x \in X,$ we have that for all $\varepsilon \in ...
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adapting proof to show interval connected

I've read and understand the proof I've attached here proving [a,b] is connected. The notes then say that is can easily be adapted to open, half open, and unbounded intervals. How would I adapt it for ...
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Metric that satisfies every property but symmetry

Suppose that we have a function $d:M\times M\to \mathbb{R}$ which satisfies every property of a metric except we have that $d(x,y)\neq d(y,x)$. Is there any interesting theories that arise from this ...
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Is the Cartesian product of finitely many metric spaces also a metric? If so, what about completeness?

Let $n$ be a positive integer, and let $p$ be a real number such that $p \geq 1$. Let $(X_1, d_1), \ldots, (X_n, d_n)$ be metric spaces, and let the set $X$ be given by $$X \colon= \Pi_{k=1}^n = X_1 ...
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a weak notion of flow in a metric space

I am seeing the definition of flow in a metric space : $f:M\times \mathbb{R}\rightarrow M$ is one flow if $M$ is metric space, $f$ is continuous and $f(x,t+s)=f(f(x,t),s)$ Note that the condition is ...
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Hausdorff dimension of F and f(F)

We have F being a subset of R, [-1,1], while f:R->R, where f(x)=x^2. What's the Hausdorff dimension of F and f(F)? I think the dim(F)=2(length) and dim(f(F))=1, is it correct? Thanks,
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Why does this proof fail?

I'm reading some notes on topology, and the notes' author is trying to raise motivation to consider compactness by providing a theorem whose proof is built intentionally wrong, but I don't agree with ...
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Does every non-singleton connected metric space $X$ contains a connected subset (with more than one point) which is not homeomorphic with $X$?

Does every non-singleton connected metric space $X$ contains a connected subset (with more than one point) which is not homeomorphic with $X$ ? Also ; does every connected metric space $X$ contains ...