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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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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37 views

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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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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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 ...
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Show that [0, 1) with the induced topology from R is a Polish space.

It's easy to see that the space is separable because $Q \cap [a,b)$ is a countably dense subset of $[a,b)$, but I can't figure out a way to show that it's completely metrizable. I know this means ...
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Showing a metric space is not complete.

Consider the metric space $$B = \{ f \in C[0,1] : \int_a^b \left| f(x) \right| dx \leq 1\},$$ where $d(f,g) = \int_0^1 \left| f(x) - g(x) \right|dx$. I'm trying to show that this metric space is not ...
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Questions about Proof that Cartesian Product of Open Sets is an Open Subset

I'm trying to understand the proof that: The cartesian product $A_1\times \cdots\times A_n$ of open subsets $A_i\subset M_i$ is an open subset of $M=M_1\times\cdots\times M_n$. It follows ...
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Does the Hausdorff property hold on closed subsets of $\mathbb{R}^n?$

I am trying to prove that given disjoint closed $A,B\subseteq \mathbb{R}^n$, there exist disjoint open $U,V$ containing $A,B$ respectively. In other words that we can take the Hausdorff property to ...
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homeomorphism from interval $[a,b]$ to $[0,1]\subset \mathbb{R}$

I need to show that every interval $[a,b]$ is homeomorph to $[0,1]\subset \mathbb{R}$. I've found this answer but it only deals with open sets, and I need an answer that deals with closed sets.
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What to do after defining a metric on a set? [closed]

Given a finite set $M$ of binary sequences of length 6: $$ M=\{\{1,0,1,0,0,1\},\{1,0,0,0,1,1\},...\} $$ Let's define a metric (Levenshtein distance) on $M$, which makes it a metric space. That's ...
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prove that $d$ is a metric.

Let $E=\{0,1\}^\mathbb{N}$, and $d: E\to \mathbb{R}$, defined by $d(x,x)=0$ and $$d(x,y)= 2^{-\min \left\{k\in \mathbb{N}\mid x_k \neq y_k\right\}}$$. For all $x=(x_k)_k,y=(y_k)_k \in E$, prove that $...
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Define a metric for an annulus, which makes it seem like the curved wall of a cylinder.

Can anybody please help me in understanding this question?
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Sequence of partial sums of e in Q is a Cauchy sequence.

Verify that $X_n= \{ \sum_{i=0}^n$ $\frac{1}{i!}$} is a Cauchy sequence in $Q$ with the Euclidean metric. I can't figure out how to find an $N$ that makes this work. I figure that $d(x_n,x_m) < \...
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Establish if $g_n (\alpha)=\int_a^b \ \alpha(x) \ \sin (nx) \ \cos(nx) $ converges uniformly

$$X=\{ \alpha:[a,b] \rightarrow \mathbb{R} \}$$ $\alpha''$ exists and it is continuous $$\exists \ K>0 \ : \forall \ x \in [a,b], \forall \alpha \in X: \\ \ \\ \rvert \alpha(x) \rvert, \rvert \...
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Natural embedding of Q with the Euclidean metric in R with the Euclidean metric is an isometric embedding.

The book I'm reading states this: The natural embedding of $Q$ with the Euclidean metric in $R$ with the Euclidean metric is an isometric embedding. What is the "natural embedding" of Q with the ...
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Banach fixed point theorem for a function $f_k(x) = k(x+1/x)$

Suppose $X = [1,\infty)$. The function $f_k(x) = k(x+\frac{1}{x})$ where $k\in(0,1)$ is a contraction on $X$, furthermore, $X$ is complete and $f:X\rightarrow X$. So all the requirements for the ...
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If for every $\epsilon>0$ there exist infinitely many $n$ such that $d(x_n,c)<\epsilon$ then $(x_n)$ has a subsequence converging to $c$.

Suppose $(X,d)$ is a metric space. I am trying to show that: If for every $\epsilon>0$ there exist infinitely many $n$ such that $d(x_n,c)<\epsilon$ then $(x_n)$ has a subsequence converging to ...
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110 views

Borel measurability of a subset of a product space

Let $X$ and $Y$ be compact metric spaces and let $\mathcal B_X$ and $\mathcal B_Y$ be their respective Borel $\sigma$-algebras. Let $\mu$ be a Borel probability measure on $X$ and let $\mathcal B^*...
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Pointwise convergence of Lipschitz functions from a compact space implies uniform convergence

Let $(f_n)$ be a sequence of $1$-Lipschitz functions from $(X, d_X)$ to $(Y,d_Y)$ where the first one is compact and the latter is complete (I am not sure if this matters). Let $f_n \to f$ pointwise. ...
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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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Does there exist a metric space of cardinality aleph-two that has a countable epsilon cover?

Does there exist a metric space $(M,d)$ such that $|M|=\aleph_2$ but there exists a countable $\epsilon$-cover of $M$?
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26 views

Cauchy sequences are bounded

As $\{x_n\}$ is a Cauchy sequence, there exists a positive integer $N$, such that for any $n \geq N$ and $m \geq N$, $d(x_n,x_m) \lt 1$; that is, $|x_n-x_m| \lt 1$. Put $M = |x_1| + |x_2| + |x_3| + .....