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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Connected metric spaces with disjoint open balls

Let $X$ be the $S^1$ or a connected subset thereof, endowed with the standard metric. Then every open set $U\subseteq X$ is a disjoint union of open arcs, hence a disjoint union of open balls. Are ...
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An elementary question about real plane metrics

Given the metric $d_p$ on the real plane, i.e., $ d_p(x,y)=[|x_1-y_1|^p+|x_2-y_2|^p]^{1/p}$ For which values of $p \geq 1$ is it true that the following set is the usual line segment in the real ...
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Hausdorff distance on power sets

Consider a general metric space $(S,d)$, with $d$ a $1$-bounded metric, and let $X,Y \subset S$ be two closed subsets of $S$. Notice that $X$ and $Y$ are not compact. Let $\mathcal{P}(X)$ denote the ...
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Please help me in this problem with some example. [on hold]

The closure of a subset of a metric space is the set of points whose distance from the set is ?
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When are the distance between points and sets well-defined?

Let $G$ be an open subset of $\mathbb C$. I would like to prove that this set $\{z\in G; d(z,\mathbb C-G)\ge 1/n\}$, where $n\in \mathbb R$, is well-defined. In another words, I would like to know if ...
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for $X\subset \mathbb{R}$, $\mathbb{R} = int X\cup Int(\mathbb{R}-X)\cup \partial X$

I need to prove: for $X\subset \mathbb{R}$, $$\mathbb{R} = int X\cup Int(\mathbb{R}-X)\cup \partial X$$ The problem is that all the proofs I've found are for metric spaces, not $\mathbb{R}$ itself, ...
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$\overline{X\cap Y}\subset \overline{X}\cap\overline{Y}$ for real numbers, case when $\overline{X\cap Y}\neq \overline{X}\cap\overline{Y}$

My proof for this is similar to this one, but I can't find an example such that $\overline{X\cap Y}\neq \overline{X}\cap\overline{Y}$ for the real numbers.
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Together with the algebra of cardinal numbers, is there analysis of cardinal numbers? [on hold]

Let $C$ be the collection of all cardinal numbers. Is there any norm, inner-product, metric (other than discrete metric), topology(other than discrete, co-finite topology) on $C$, which is very ...
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Does the function $d(x,y)= \frac{\lvert x-y\rvert} {1+{\lvert x-y\rvert}}$ define a metric on $\mathbb{R}^n?$

Does the function $d: \mathbb{R}^n \times \mathbb{R}^n \to \mathbb{R}$ given by: $$d(x,y)= \frac{\lvert x-y\rvert} {1+{\lvert x-y\rvert}}$$ define a metric on $\mathbb{R}^n?$ How do you go about ...
0
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Banach fixed-point theorem for a recursive functional equation

I was asked to prove that the functional equation $$ x(t) = \begin{cases} \frac{1}{2} x(3t) + \frac{1}{2},& 0 \leq t \leq \frac{1}{3}\\\ f(t), & \frac{1}{3} < t\leq \frac{2}{3}\\\ ...
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Equivalence of precompactness and completely boundedness.

Definitions: The set $A\subset X$ is called completely bounded if $\forall \epsilon >0 \ \exists x_1,...,x_k \in A$ s.t. $A \subset \bigcup_{i=1}^k B(x_i,\epsilon)$. The set $A$ is called ...
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Proof that a discrete space (with more than 1 element) is not connected

I'm reading this proof that says that a non-trivial discrete space is not connected. I understood that the proof works because it separated the discrete set into a singleton ${x}$ and its ...
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Question regarding metric spaces and union of balls

If $X$ is a compact space and $\epsilon > 0$, I want to show that there exists $n$ point $x_1, x_2, ... x_n$ such that $\bigcup_{i=1}^n B(x_i, \epsilon) = X$ I am not sure how to do that. ...
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1answer
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Distances in a metric space

If $X$ is a metric space and $x_0\in X$. Let $x$ and $x'$ be any points of $X$. I want to unerstand why the following inequality is correct: $d(x,x_0)-d(x',x_0) \leq d(x,x')$ I understand that if we ...
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Cartesian product of compact sets is compact

Prove that if two sets $A$ and $B$ are compact then so is their Cartesian product $A \times B = \{(a,b): a \in A, b\in B\}$. The hint is to use Bolzano Weiertrass theorem and an argument of sequence ...
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A-noncompact, Does there **always** exist a continuous function $f: A \to \mathbb R$ which is bounded but does not assume extreme values?

It's well known that if $ A \subset \mathbb R$ is compact then every continuous function $f:A \to \mathbb R$ is bounded and assume extreme values .So the obvious question is: Given any non compact ...
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(Lipschitz/Uniform) Continuity of a map [on hold]

I need help with proving / disproving something. I'm really bad at TeX so maybe someone can help me formatting. Look at the map $$Φ: (C([0,1]), \mathbb R), ||·||_{sup}) ~ \to ~(\mathbb R, |·|); ...
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If $S \subset X$ does a subsequence of $S$ converge in $S$ or in $X$?

Let $X$ be a metric space and let $S\subset X$ be a compact space. By definition, $S$ is compact implies that for all sequences $(x_n)$ of $S$, there exists a subsequence $(x_{n_{\alpha}})$ that ...
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Proof/disprove contunuity of a map [duplicate]

I need help with proving / disproving something: Look at the map $$\Phi: (C([0,1], \mathbb R), ||\cdot||_{\infty}) \to (\mathbb R, |\cdot|); \,\,\,\,\,\,\Phi(u) := \int_0^1 u²(t) dt$$ a) ...
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Prove that a linear mapping between vector spaces is an open mapping iff

Let $(N,|| \ ||)$ and $(N_1,||\ ||_1)$ be normed vector spaces and $f$ a linear mapping of $N$ into $N_1$. Prove that $f$ is an open mapping if and only if $\forall$ $n \in \Bbb N $, $B_r(0) ...
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Metric spaces - $(0, 1)$ and $\mathbb{R}$ are not isometric

I'm looking for alternative solutions to what I currently have for the sake of self studying to the following: Show that $(0,1)$ and $\mathbb{R}$ are not isometric, where both sets are equipped ...
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Definition of bounded in a metric space - confirmation

Is the following definition of a bounded metric space correct? $(M,d)$ is bounded if $\exists a \in M, r > 0$ such that $M = B(a,r)$. Looking around on the internet I instead see $M \subset ...
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1answer
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How does the union in $\Bbb R^2$ look like?

Let $$\bigcup_{n=1}^\infty [\frac{1}{n+1},\frac{1}n) \times (0,n) \subset \Bbb R^2 $$ be a subset. I need to know how this look like in $\Bbb R^2 $ for my homework, but I'm not sure. First I thought ...
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May a 'ball' that has been 'cut off' still be called a 'ball'?

Consider the metric subspace $[0,1] \subseteq \mathbb{R}$ with the metric defined in the usual sense, and the ball $B(0,1)$, defined to be the ball centred at $x=0$ with radius $1$. Now since only ...
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1answer
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Can you give me an example of a function that is either upper OR lower quasi-continuous but not both?

A function $f: X \rightarrow \mathbb{R}$ is said to be upper (lower) quasi-continuous at $x \in X$ if for each $\epsilon >0$ and for each neighbourhood $U$ of $x$ there is a non-empty open set $G ...
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How do I show that a contraction mapping in a metric space is continuous?

I start out by letting $V$ be an arbitrary open set in $X$. Then $$ f^{-1}(V) = \{x\in X\mid f(x) \in B_\epsilon(f(a))\}. $$ This can be re-written as: $$ f^{-1}(V) = \{x\in X\mid d(f(a), f(x)) < ...
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Upper and lower bound for the separation of two trajectories of a dynamical system

Consider trajectories $x(n)$ and $y(n)$ of the tent map, starting from initial conditions $x(0)$ and $y(0)$. Then the distance $δ$ between the trajectiories is: $δ = |x(n) - y(n)| = \exp (λ n)|x(0) - ...
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Show that $\bar{A}=\{x \in M | d(x,a)=0\}$

Let $(M,d)$ be a metric space. Let $A$ be an arbitary subset of $M$ and let $x$ be an arbitary point. Define $d(x,A)=\inf \{d(x,y)\mid y \in A\}$. Show that $\bar{A}= \{x \in M \mid d(x,A)=0\}$ How ...
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Example of a locally compact metric space which is $\sigma$-compact but not proper

Let $(X,d)$ be a locally compact metric space. Then it is known that $X$ is separable if and only if it is $\sigma$-compact (i.e. it can be written as a countable union of compact sets). Moreover, ...
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Proving that the subset is the set itself

I am trying to prove the following property, which seems fairly intuitive, at least in $\mathbb{R}^n$. Let $(X,d)$ be a compact metric space where $Y \subseteq X$ arbitrary. Prove that if there ...
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Show that $f$ is uniformly continuous $\iff$ $\bar d(f(x_n),f(y_n)) \to 0$ for all sequences $\{x_n\}, \{y_n\}$ in $X$ with $d(x_n, y_n) \to 0$.

Suppose $(X, d), (Y,\bar d)$ are metric spaces, $f:X \longrightarrow Y$. Show that $f$ is uniformly continuous $\iff$ $\bar d(f(x_n),f(y_n)) \to 0$ for all sequences $\{x_n\}, \{y_n\}$ in $X$ with ...
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If $A$ is a component in $(X,d_1) \iff$ $A$ is a component in $(X,d_2)$ then is $(X,d_1)$ homeomorphic to $(X,d_2)$?

Suppose that $A$ is a component in $(X,d_1) \iff$ $A$ is a component in $(X,d_2)$. Is it always the case that $(X,d_1)$ is homeomorphic to $(X,d_2)$? I have been trying to find a counter example, but ...
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$M$ connected $\iff$ $M$ and $\emptyset$ are the only subsets of $M$ open and closed at the same time

I'm trying to understand this proof that: $M$ connected $\iff$ $M$ and $\emptyset$ are the only subsets of $M$ open and closed at the same time Which is: If $M=A\cup B$ is a separation, then $A$ ...
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$f$ continuous $\iff f(B(a,\delta))\subset B(f(a),\epsilon)$

My book says that when $f$ is continuous, we have that $\forall \epsilon>0$, there exists $\delta>0$ such that: $d(x,a)<\delta \implies d(f(x),f(a))<\epsilon$ Then, my book says that ...
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$\partial A$, when $A=\{x\in M: f(x)>0\}$ is the set $\{x\in M: f(x) = 0\}$

I have a question about the proof of this fact: $\partial A$, when $A=\{x\in M: f(x)>0\}$ is the set $\{x\in M: f(x) = 0\}$ The proof says the following: $$A = f^{-1}((0,+\infty))$$ Since ...
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Why is $l^\infty$ not separable?

My functional analysis textbook says "The metric space $l^\infty$ is not separable." The metric defined between two sequences $\{a_1,a_2,a_3\dots\}$ and $\{b_1,b_2,b_3,\dots\}$ is ...
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Euclidean distance between $x\in\mathbb{R}$ and $\{x\in\mathbb{R} \mid f(x)=0\}$ [closed]

Is there a generic formula to calculate the distance between an arbitrary real number $x\in\mathbb{R}$ and $$\{x\in {\mathbb{R}}\mid f(x)=0\}$$ where we have little information about $f$? In fact, my ...
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$F=\{x\in M: f_\lambda(x)\ge 0, \forall \lambda\in L\}$ is a closed subset of $M$ [closed]

I want to discuss this proof that: Let $f_\lambda$ be a family of continuous functions, then: $F=\{x\in M: f_\lambda(x)\ge 0, \forall \lambda\in L\}$ is a closed subset of $M$: Since every ...
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Homeomorphism between $\mathbb{R^2}$ and $S^2-N$, the sphere without its north pole

How would one approach the following problem? Write down a homeomorphism and its inverse from $\mathbb{R^2}$ to the sphere $S^2-N$ without its north pole So I need a function $f(x,y) : ...
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1answer
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Retracts and dense $G_{\delta}$ sets

Suppose $X$ is a complete separable metric space and $Y\subset X$ is a retract of $X$ with retraction $r\colon X\longrightarrow Y$. I am interested in the following question: Given a dense ...
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For $f:M\to N$ to be continuous its sufficient that $x_n\to a\implies f(x_n)_n$ is convergent in N

In order to prove: For $f:M\to N$ to be continuous its sufficient that $x_n\to a\implies f(x_n)_n$ is convergent in N I'm supposing that $x_n$ is convergent, that is: $$\forall \epsilon>0, ...
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Graph of a continuous function $f:M\to N$ is a closed subset of $M\times N$

I need to prove that the graph of a continuous function $f:M\to N$ is a closed subset of $M\times N$. $N$ is a metric space. I think I'm supposed to use this result. So, that's what I did: $Graph(f) ...
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$M=A\cup B$, $f|_A$ and $f|_B$ are continuous, then $f$ is continuous in $A\cap B$

In order to prove: $M=A\cup B$, $f|_A$ and $f|_B$ are continuous, then $f$ is continuous in $A\cap B$ does it suffice to prove: for $a\in A\cap B$: since $f|_A$ is continuous, then $\forall ...
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Continuity in terms of interior of preimage and preimage of interior

Let $f$ be a map between metrix spaces $X,Y$. In order to prove: $f$ is continuous $\iff$ $f^{-1}(\operatorname{Int} Y)\subset \operatorname{Int}(f^{-1}(Y))$ I did: $\rightarrow$ Suppose $x\in ...
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Non-compact subsets of a metric space $(X,d)$.

I'm trying to come up with an example of a metric space $(X,d)$ such that a subset $A \subset X$ is not compact, but is closed and bounded. Essentially I want to find an example that shows that a ...
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$(f(x_n))_{n\in \mathbb{N}}$ converges for all continuous $f$. Does $(x_n)_{n \in\mathbb{N}}$ converge?

Let $(S, d)$ be a metric space and $(x_n)_{n\in \mathbb{N}}$ a sequence in $S$. If $(f(x_n))_{n\in \mathbb{N}}$ converges for all continuous $f:S\to\mathbb{R},$ does it follow that $(x_n)_{n\in ...
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Proving that sums of convergent sequences are complete metric spaces

Let $L_1$ be the set of all sequences of real numbers $$x = (x_1,x_2,..., x_n, ...) $$ with the property that $\sum_{n=1}^\infty |x_n|$ is convergent. If we define $$d_1(x,y) = \sum_{n=1}^\infty ...
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Prove $\bar{A}\setminus \bar{B} \subset \overline{A\setminus B }$

Here is my approach so far Let $A$ and $B$ be subsets of the metric space $(M,d)$ My thoughts on how to prove it, is to choose an element $x$ from $\bar{A}\setminus \bar{B}$ and show it exists in ...
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Proving equivalence of statements on continuity between metric spaces

On page 228 of Mícheál Ó Searcóid's Metric Spaces (2007), he writes Criteria for Comparability of Metrics Suppose $X$ is a set and $d$ and $e$ are metrics on $X$. Then the following ...
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1answer
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Definition of a separable metric space

The book I'm reading doesn't explicitly give a definition of separable metric spaces. The only type of separability definition I know that a separable topological space is one that has a countable ...