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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$\overline{X\cap Y}\subset \overline{X}\cap\overline{Y}$ 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? [closed]

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 useful?...
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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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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}\\\ \frac{1}{...
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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 ...
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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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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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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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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 B(...
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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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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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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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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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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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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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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 $d(...
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$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 $(0,+\...
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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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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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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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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) : \mathbb{...
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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 f^{-1}...
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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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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 ...
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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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if $K\subset G$ there is an $\epsilon>0$ such tht $K_\epsilon\subset G$

Let $(M, d)$ be a metric space and $K$ a compact set and $G$ an open set such that $K\subset G$ . Is it true that there exists an $\epsilon>0$ such that $K\subset K_\epsilon \subset G$? Notation: ...
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How does the compactness property help us show a subset $A$ of a metric space $X$ is closed?

We have a compact subset $A$ of a metric space $X$ and we want to show that this implies that $A$ is closed. Let $y \in A$ and $y \in A^c$. For each $y \in A$, we can take open neighbourhoods $U_y$ ...
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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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Constructing a metric topology that is the same as the standard topology

Any help on this problem would be greatly appreciated. thanks! $\textbf{Definition:}$ Let $\tau$ be the collection of subsets of $\mathbb{R}^n$ with the following property: $\forall x \in U,\; \...
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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 |x_n-...
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$K$ is compact and $x\in X$ but $x\notin K$. Show $\exists G_1,G_2$ open in $(X,d)$ s.t. $x\in G_1$ and $K\subseteq G_2$

Suppose $K$ is a compact subset of a metric space $(X,d)$ and $x \in X$ but $x\notin K$. Show that there exist two disjoint open sets of $G_1$ and $G_2$ of $X$ such that $x\in G_1$ and $K\subseteq G_2$...
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Let $f:A\to N$, show that if there exists $\lim_{x\to a}f(x)$ we have $b\in \overline{f(A)}$

I have the following exercise: Let $f:A\to N$, show that if there exists $\lim_{x\to a}f(x)$ we have $b\in \overline{f(A)}$ I don't know what $b$ is meant to be, there's a typo in this exercise. I ...
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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 complementar....
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$A$ is an open subset of $M$ $\iff$ ($x_n\to a\implies x_n\in A$ for large $n$)

My definition of an open subset $A$ of $M$ is the one that for every $x\in A$, there is an open ball contained in $A$. Now, suppose that $x_n\to a$. By definition, $\forall \epsilon>0$ there exists ...
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$0\le d(x_n, a)<\frac{1}{n}\implies \lim x_n = a$

I need to prove the following: $$0\le d(x_n, a)<\frac{1}{n}\implies \lim x_n = a$$ It looks pretty intuitive since I can make $\frac{1}{n}$ as small as I want, thusk making $a$ as close as to $...
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61 views

Show completeness of metric subspace

I have problems solving the following 2 problems: Given is the metric $d:\Bbb R\times\Bbb R\to[0,\infty[$ with $$d(x,y):=|\arctan(x)-\arctan(y)|\;.$$ a) Show that the metric subspace $\...
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Questions about proof of $\lim x_n = a, \lim y_n = b\implies \lim x_n+y_n = a+b$ in a normed vector space

I need to prove that, in a normed vector space $E$, we have: $$\lim x_n = a, \lim y_n = b\implies \lim (x_n+y_n) = a+b$$ and: $$\lim\lambda_n = \lambda, \lim x_n = a \implies \lim \lambda_n\cdot ...
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Example of an uncountable metric space where every point is isolated

I was trying to come up with an example of an uncountable metric space all of whose points are isolated. I've had difficulty thinking of one, has anyone got any nice examples? Just in case: ...
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Show that $\{f_n(x) \}_{n \in \mathbb{N}}$ doesnt converge in M.

Let $n \in \mathbb{N}, f_n(x)=e^{-(x-n)^2}$ and $g(x)=\Bigg\{\begin{array}{ll} 1 & x=0 \\ \frac{1-e^{-x^2}}{x^2} & x \neq 0 \\ \end{array} $ You can assume that g is continuous ...
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2answers
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If two sequences are Cauchy, then d(sequence_1, sequence_2) is cauchy in R

The question says this: If $(X,d)$ is a metric space and $\{x_n\}$ and $\{y_n\}$ are Cauchy sequences, prove that $\{d(x_n,y_n)\}$ is a Cauchy sequence in $R$. I see that I would have to show that $...
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If $X$ is totally bounded then every sequence contains a Cauchy subsequence

I attempted the proof, I just want to see if it is correct: Suppose $X$ is totally bounded and $(x_n)$ is a sequence in $X$. Then $(x_n)$ has a subsequence contained in a ball of radius $1/2$. This ...