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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Proof of theorem $20.5$ Munkres Topology

First the metric on $\mathbb R^{\omega}$ is defined as $$D(x,y)=sup\left\{ {\bar d(x_i,y_i)}\over i \right\}$$ where $\bar d(x,y)=\min\{d(x,y),1\}$ and $\bar d$ is the Euclidean metric on $\mathbb ...
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Regarding the Manhattan metric and the projections of a closed ball.

In $R^2$ If I have the closed ball $\bar{B}(0,1)$ in the Manhattan metric and I take the set $A_x = \{ y \in R| (x,y) \in \bar{B}(0,1) \} $ and $P_A = \{ x \in R | (x,y) \in \bar{B}(0,1) \}$ (the ...
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Showing that If $A$ and $B$ are closed disjoint subsets of a metric space then there exists disjoint open sets containing $A$ and $B$.

Suppose $A,B$ are closed subsets of a metric space $(X,d)$. I am trying to show that there exist disjoint open sets $U,V$ such that $U$ contains $A$ and $V$ contains $B$. I managed to find an open ...
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“Accumulation” points of a convergent net

Let $(X,d)$ be a metric space, $D$ a directed set and $\phi :D \rightarrow X$ a net converging to some $x_0 \in X$. Can there be an increasing sequence $\{d_n\} \subset D$ s.t. $\{\phi(d_n)\}$ ...
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Show that a sequence converges in on metric iff the sequence converges in another metric.

Let $\delta(x,y) = \bigg | \frac{1}{x}-\frac{1}{y} \bigg|$ and $d$ is the usual euclidean metric. Show that $(x_n)$ converges to $a$ using $\delta$ iff it converges to $a$ using $d$. My attempt: ...
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Metric space $M = M_1\times \cdots M_n$ with metric $d''(z,z') = \max \{d_i(x_i,x_i'), i\in \{1,\cdots,n\}\}$. Showing triangular inequality.

I have the following metric space: $M = M_1\times \cdots \times M_n$ with metric $d''(z,z') = \max \{d_i(x_i,x_i'), i\in \{1,\cdots,n\}\}$ where $d_i$ is the metric for each $M_i$, and ...
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Pick $b\notin B[a,r]$ show that there exists $s>0$ such that $B[a,r]\cap B[b,s]$ is empty

I need to solve this question: Pick $b\notin B[a,r]$ show that there exists $s>0$ such that $B[a,r]\cap B[b,s]$ is empty My idea is to suppose a point $p$ in the intersection. Then ...
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Polar coordinates in taxicab geometry

We know that in euclidean $\mathbb{R}^2$ space polar coordinates are defined by $$r = \sqrt{x^2 + y^2}$$ $$\theta = \arctan\frac{y}{x}\text{.}$$ Now, geometrically we can think of it as of point, ...
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$\Delta = \{(x,x), x\in M\}\subset M\times M$. Show that if $z\in M\times M - \Delta$ then there is a ball with center $z$, disjoint from $\Delta$

I need to show this: $\Delta = \{(x,x), x\in M\}\subset M\times M$. Show that if $z\in M\times M - \Delta$ then there is a ball with center $z$, disjoint from $\Delta$ I need to use the metric ...
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metric spaces - basic inequality

Let $(\Omega, d)$ be a metric space. I have to show that $ d(\alpha ,\beta) \ge | d(\alpha, \theta) - d(\theta, \beta)|$ for every $\alpha, \beta, \theta \in \Omega.$ Starting with the triangle ...
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If G is an open subset of a metric space, is it true that $\text{int}(\bar{G}) = G$?

If G is an open subset of a metric space, is it true that $\text{int}(\bar{G}) = G$? I have not been able to find a counter example or a proof. Any hints?
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Looking for an example of an infinite metric space $X$ such that there exist a continuous bijection $f: X \to X$ which is not a homeomorphism

I am looking for an example of an infinite metric space $X$ such that there exist a continuous bijection $f: X \to X$ which is not a homeomorphism . Please help . Thanks in advance .
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How to proof an open set and its complement contained in a metric space with infinite element are both infinite? [duplicate]

Let X be a set containing infinitely many elements, and let d be a metric on X. Prove that X contains an open set U such that U and its complement $U^c$ = X\U are both infinite.
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How does the “arc tangent metric” $d(x,y) = | \arctan(x) - \arctan(y)| $ work?

I see there are some counterexamples and so forth in metric spaces regarding the metric $$d(x,y) = | \arctan(x) - \arctan(y)| $$ But honestly I have no intuition as to how it works For example, in ...
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Is it always true that ${c(\bar A)} = \overline{c({A})}$?

Suppose $(X,d)$ is a metric space. If $A$ is a closed subset of X then ${c(\bar A)} = \overline{c({A})}$ where $c$ is the complement of the set and $A$ is a subset of the metric space. I think ...
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Let $f: X \subset \mathbb{R}^n\to \mathbb{R}^m$. Then, $f$ is uniformly continuous if, and only if, for every sequence…

Let $f: X \subset \mathbb{R}^n\to $. Then, $f$ is uniformly continuous if, and only if, for every sequence, $(x_n)_{n\in\mathbb{N}}$, $(y_n)_{n\in\mathbb{N}}$ such that $d(x_n,y_n) \to 0$, then ...
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Open ball cartesian product on metric space: $B(a,r) = B(a_1, r)\times \cdots \times\ B(a_n,r)$

I need to prove that $$B(a,r) = B(a_1, r)\times \cdots \times B(a_n,r)$$ in $M=M_1\times\cdots \times M_n$ where $M_i$ is a metric space and the metric is $d''(z,z') = \max\{d_i(x_i,y_i), i \in ...
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Is the set $\{\big(x,\sin(1/x)\big):x\ne 0 \}$ connected in usual metric of $\mathbb R^2$? [duplicate]

Is the set $\{\big(x,\sin(1/x)\big):x\ne 0 \}$ connected in usual metric of $\mathbb R^2$ ? I tried writing it as a union of two connected sets , or otherwise as a union of two disjoint non-empty ...
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Interior and Exterior Points in C[0,1] with Supremum Metric

Let $C[0,1]$ be the set of real, continuous functions on the set $[0,1]$ with the metric \begin{equation} d(f,g)=sup_{x\in [0,1]} |f(x) - g(x)|. \end{equation} Consider the set $C$ of constant ...
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Sequences in metric spaces.

Given , $X= l_p (p\geq 1)$ , and let $d(x,y) = ( \sum_{k=1}^{\infty} |x_k - y_k |^{p})^{\frac{1}{p}}$ where $x= \{x_k\}_{k\geq 1}$ and $y= \{y_k\}_{k\geq 1}$ are in $l_p$. Let $\{x^{(n)}\}_{n \geq ...
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Metric spaces whose open sets form a $\sigma$-algebra.

I have the following question Characterise the metric spaces whose open sets form a $\sigma$-algebra. I apologise if this seems like too basic of a question to ask here, but seeing as English is ...
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32 views

Find $\text{dist}(i,A)$ where $A = \{z\in\mathbb{C}:|z-(1-i)|<1\}$

I am trying to find $\text{dist}(i,A)$ where $A = \{z\in\mathbb{C}:d(z,(1-i))<1\}$ where $d$ is the usual metric on $\mathbb{C}$, that is d(x,y) = |y-x|. I know that the set A in the complex plane ...
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Completion of polynomial space with max norm

Let's start with space of finite polynomials over $\mathbb{R}$. Weierstrass theorem says it's dense in $C[0;1]$ by norm $\|P(t)\|=\max|P(t)|, t \in [0,1]$. So the completion of this space is $C[0,1]$. ...
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Proving Corollary to Riesz's Lemma

Let $(X,\|\cdot\|)$ be a normed linear space and $Y \leqslant X$ be a proper subspace. If $\text{dim}(Y) < \infty$, show that there exists $x \in X$, with $\|x\| = 1$ such that $d(x,Y) = 1$. ...
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Example for converges series in the metric space

Give example for converges series in the metric space: $$ \quad\quad\quad(\mathrm {R}^n,d_{\infty}),d_\infty=\max\mid x_i-y_i\mid$$ My attempt: Let ...
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Distance inequality

Consider $H=\{ (x,y) \in {\bf R}^2\mid x$ or $y$ is an integer $\}$ If $d$ is canonical distance in ${\bf R}^2$, show that if $d(x):=d(x,H)$, (1) $$ d(x) - d(y) \leq d(x,y) $$ if $x,\ y$ are in same ...
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How do I show that the open rectangle is convex?

I have no ideia how to solve this. How to show that the open ball is convex or the open cube is, that`s easy, but how to show that the open rectangle is? (The same holds for the closed rectangle). ...
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Compact subsets of metric space with French railway metric

Let $A=\{0,1,2,...\}$ with $f$ the French railway metric that has centre $0$ and $f(a,0)=1$ for all $a\in A$ with $a\neq0$. How do I show that the metric space $(A,d)$ is complete? How do I ...
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Are $d(f,g)=\max \limits_{a\leq x \leq b} |f(x)-g(x)|$ or $d(f,g)=\min \limits_{a\leq x \leq b} |f(x)-g(x)|$ metric?

Neither $d(f,g)=\max \limits_{a\leq x \leq b} |f(x)-g(x)|$ nor $d(f,g)=\min \limits_{a\leq x \leq b} |f(x)-g(x)|$ meet the triangle equality condition to be metric. Because if you some $x$ (say $x_1$) ...
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Is distance function defined on a convex set is always convex?

I am looking for an answer to the following question: Is the distance function defined on a convex set always convex? Obviously the convex set in question is metric. In particular I am interested ...
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Show that the subspace $Y = \{ x \in \mathcal{C}[a,b] \mid x(a) = x(b) \} \subset \mathcal{C}[a,b]$ is complete

Now I understand that the function space $\mathcal{C}[a,b]$ of continuous functions from the closed interval $[a,b]$ to $\mathbb{R}$ is complete, with the metric: $ d(x, y) = \underset{t \ \in \ ...
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Show two metrics are equivalent

The definition I'm given is that two metrics $\tau, \rho$ on a set X are equivalent if there exist two numbers $c_1, c_2$ such that $c_1\tau(u,v)\leq\rho(u,v)\leq c_2\tau(u,v)$. I'm supposed to show ...
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Accumulation points and successions in metric spaces

let $(M,d)$ be a metric space, $A\in M$. Show that a dot $a\in A$ es a dot of acumulation of A $\leftrightarrow$ $\exists$ a succession of dots $(a_n)$ with $a_n\in A$ such that $a_n$ tends to $a$
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Travel of the point via isometry: does it get closer to home?

Let $X$ be a compact metric space, $f\colon X\to X$ — isometry. We fix an arbitrary point $x\in X$ and consider sequence $a_n = \underbrace{f(...f}_{n\ \ \mathrm{times}}(x))$. The hypothesis is ...
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How to prove that is a Cauchy sequence

$\mathbb{R}$ is endowed with the metric $d(x,y)=|\arctan(x)-\arctan(y)|$ , i want to prove that $(x_n=n)$ is a Cauchy sequence, i see that $\lim_{p,q\rightarrow\infty} d(x_p,x_q)=0$ then $(x_n)$ is a ...
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A sequence with no Cauchy subsequence

let $E=\mathcal{C}([0,\pi],\mathbb{R})$ and $$d(f,g)=\sqrt{\int_0^{\pi} (f(x)-g(x))^2 dx}, ~\forall f,g\in E$$ Hello, please How to prove that $f_n(x)=\sin(nx), n\in \mathbb{N}$ has no a Cauchy ...
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$X$ be Banach space , $T \in \mathcal B(X)$ be an open map , $Y$ be a closed linear subspace of $X$ ; is the restriction of $T$ on $Y$ an open map?

Let $X$ be a Banach space , let $T$ be a continuous open linear map from $X$ to $X$ , let $Y$ be a closed linear subspace of $X$ , then is $T_o$ , the restriction of $T$ on $Y$ , is an open map ? ...
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A sequentially compact metric space is bounded. Help me fix this proof.

I know this is usually done by contradiction but I'm trying out something a bit different: Let $\mathbb{X}$ be a sequentially compact metric space. Let $(s_n)$ be a sequence in the metric space such ...
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Subspace of a seperable space is not neccessarily seperable in topology space [duplicate]

Let $(X,d)$ be seperable metric space, then $(Y,d_1)$ where $Y \subset X$ and $d_1$ is induced metric on a subspace $Y$. Then $(Y,d_1)$ is a seperable metric space, too. I think it is quite easy to ...
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how do i prove that these sets are the same? [duplicate]

Let (M,d) be a metric space, with $E\subset M$: And for this one; should i take distance from an element to itself or something? prove that: b)$\{ x: d(x,E)=0\} = \overline E$
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Prove that $f(x)$ is continuous on $M$ [duplicate]

Let $(M,d)$ be a metric space, $A\subset M$. Prove that $$f(x)=d(x,A)=\inf_{y\in A}d(x,y)$$ is continuous on $M$. I have tried the following Let $\epsilon>0$, $x_0\in M$. I have to find ...
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Bounded sequence in metric space has convergent subsequence

Exercise 12.5.11 in Tao's Analysis 2. Let (X,d) be a metric space. Claim: For every open cover of (X,d) there is a finite subcover $\implies$ X compact. Proof: If X is not compact, then there is a ...
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90 views

Regarding “stronger” norms

Let $X$ be a normed linear space. Show that a norm $\|\cdot\|_{1}$ is stronger than a norm $\|\cdot\|_{2}$ if and only if for any sequence $\{x_{n}\} \subset X$, $\|x_{n}\|_{1} \to 0$ always ...
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Does there exist an injective function $f:\mathbb R^2 \to \mathbb R$ such that $f$ is continuous in one of the variables $x$ or $y$?

Does there exist an injective function $f:\mathbb R^2 \to \mathbb R$ such that $f$ is continuous in one of the variables $x$ or $y$ ? I only know that such an injection cannot be continuous in each ...
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Counterexample of intersection characterization of compactness

Given metric space ($X,d$), let {$S_1, S_2,...$} be a set of non-empty sets where $S_1 \supseteq S_2...$, then if $X$ is compact and the $S_t$ are closed then $ \cap_tS_t$ is not empty. In $\Bbb R$, ...
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How to call a mathematical space “$(\mathcal S, f)$” consisting of set $\mathcal S$ and function $f : \mathcal{S \times S} \rightarrow \mathbb R$?

Is there a specific standard name for a mathematical space "$(\mathcal S, f)$" consisting of a set $\mathcal S$ and a function $f : \mathcal{S \times S} \rightarrow \mathbb R$; perhaps together with ...
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49 views

Showing that $d(x,y) = \mbox{min}\{1,|x-y|\}$ is a metric

I need to show that $$d(x,y) = \mbox{min}\{1,|x-y|\}$$ is a metric. I started proving that $$d(x,y)\ge 0$$ because $|x-y|$ is a metric, so $|x-y|\ge 0$ and that $1\ge 0$. Now, we know that ...
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2answers
67 views

Does there exist a continuous injection from $[0,1)$ to $(-1,1)$?

Does there exist a continuous injective or surjective function from $[0,1)$ to $(-1,1)$ ? I know there is no continuous bijection from $[0,1)$ to $(-1,1)$ , but am stuck with only injective continuous ...
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53 views

Not locally compact why?

let $E=\mathcal{C}([0,\pi],\mathbb{R})$ and $$d(f,g)=\sqrt{\int_0^{\pi} (f(x)-g(x))^2 dx}, ~\forall f,g\in E$$ How to prove that $E$ is not locally compact ? i proved that $f_n(x)=\sin(nx), n\in ...
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Does a (semi)metric exist on a vector space (or other) that can be expressed as a small polynomial with a large number number of equidistant points?

Suppose we have some "large" number of categories, say $N \geq 2^{20}$, and $M$ observations, each having one of the $N$ categories. The description for each category is perfect; perhaps we can ...