Metric spaces are sets on which you can measure the "distance" between any two points. The distance measurement is generally required to be symmetric (so distance from $A$ to $B$ is the same as distance from $B$ to $A$), positive for two distinct points, and obeying the triangle inequality.

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Prove with complete metric space

Let $(X,d)$ be a metric space such that, for every $x \in X$ and $r>0$, the closed ball $$\overline{B}(x,r)=\{y \in X:d(x,y)\leq r\}$$ is compact. Prove that $X$ is complete. My attempt: Let ...
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34 views

Definition of continuity in practice

In general I have a problem to recognise if a function is continuous or not. I simply don't know where I should start to actually see it. Here there is an example of my problem that I found in a ...
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23 views

Finding a finite cover. [on hold]

Let $(X,d)$ be a metric space and $U$ be a bounded open connected subset of $X$. Show that it is possible to find a finite collection of closed sets $\{B_i\}$ which cover the closure of $U$ and the ...
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If $A$ be a connected set in a metric space and $B\subseteq A^‎{'}$, which is the connected set? [on hold]

Let $A$ be a connected set in a metric space and $B\subseteq A^‎{'}$. which the follow set is the connected set? $A^{\circ}$ $\overline{A}-B$ $\overline{A}-A$ $A\cup B$
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24 views

Which the follow property is correct? [on hold]

Let $X$ be a metric space and $E‎\subseteq‎ X$. Which the follow property is correct? for every $E‎\subseteq‎ X$,$\overline{E^\circ}=\overline{E}$ for every $E‎\subseteq‎ ...
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1answer
38 views

Is there exist a ball with lesser radius than another ball that contains it?

If $B_1$ and $B_2$ are two balls in metric space $X$ with radius $r_1$ and $r_2$, respectively and $B_1‎\subseteq‎B_2$,Is it possible that $r_1>r_2$ ? I think, it can occure in discrete metric ...
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48 views

If $E_i$ is open show $\cap E_i$ is open

Question If $E_i \subseteq \mathbb{R}^p$ is open for all $i=1,2 \dots, n$. Show that $\displaystyle \bigcap_{i=1} ^n E_i$ is open. My attempt: Let $x \in \displaystyle \bigcap_{i =1}^n ...
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1answer
19 views

If $A$ , $B$ are dense in the metric space $X$ then,…

Let $X$ is a metric space and $A$ and $B$ are two dense subset in $X$. Which is correct? if $A$ is open, $A‎ \cap‎‎B$ is dense in $X$ if $A$ is closed in $X$, $A‎ \cap‎‎B=\emptyset$ $(A-B)\cup(B-A)$ ...
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22 views

On quasi-equivalence of norms

Two norms $p(v),q(v)$ are equivalent if there exist two real constants $c,C$, with $c > 0$ such that for every vector $v$ in $V$, one has that: $c q(v) ≤ p(v) ≤ C q(v)$. Can we something about ...
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1answer
21 views

Path-connectedness of continuous functions

I want to prove that the metric space $C[0,1]$ with the metric $d(f,g) = sup_{x \in [0,1]} |f(x) - g(x)|$ is path-connected. I think I've done most of the proof, but I am not too sure about the ...
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28 views

Proving that the metric space $((0,\infty),d)$ is complete, with $d(x,y)=|\ln x-\ln y|$

Let $X$ denote $(0,\infty)\subseteq \mathbb{R}$, and let $d:X\times X\to \mathbb{R}$ be defined as $d(x,y)=|\ln x- \ln y|$. Show that $(X,d)$ is a complete metric space. I am taking for granted ...
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1answer
35 views

Rank as norm on matrix

Could we consider matrix rank $r$ a norm? Is other norm similar to rank $r$ possible to associate with a finite matrix? (We denote trivial valuation $|\alpha|=1,\quad\forall\alpha\in\Bbb F^*$ where ...
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1answer
23 views

How to show $\max\{|b_2-a_2|,|b_1-a_1|\}$ is a metric?

If $a=(a_1,a_2),b=(b_1,b_2),c=(c_1,c_2)\in\mathbb{R}^2$. How can I show that $\max\{|b_2-a_2|,|b_1-a_1|\}\leq\max\{|c_2-a_2|,|c_1-a_1|\}+\max\{|c_2-b_2|,|c_1-b_1|\}$
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1answer
18 views

Making $(0,1)$ complete with a metric $d$ which defines the same topology as that of the Euclidean metric

Can we define a metric $d$ on $(0,1)$ such that the topology induced by this metric is same as that of the usual Euclidean metric on this set ?
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18 views

Topologists Sine curve Geometrically

Topologists Sine curve Geometrically Consider $A=\{(x,y):0<x\leq 1;y=\sin {\dfrac{1}{x}}\}$ and $B=\{(0,y):-1\leq y\leq 1\}$. Now $X=A\cup B$ is connected. But I want to view geometrically that ...
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1answer
19 views

Proof of a corollary of the Banach Fixed Point Theorem

If $(X,d)$ is a complete metric space, and $f: X \rightarrow X$ is a continuous function, show that if $f^{N}$ is a contraction (for some $N > 0$),then $\exists! x \in X$ such that $f(x) = x$. I ...
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1answer
39 views

Let $(X,d)$ be a metric space and $f:X\to X$ a function, is $d(x,f(x))$ a lower semicontinous function?

So I was trying to prove that if $f$ satisfies a special property the the function $d(x,f(x))$ is lower semicontinous but then I couldnt come up with a counter example of the following statement: Let ...
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28 views

To prove that any Linear map $T : \Bbb R^m \to \Bbb R^n$ is uniformly continuous.

To prove that any Linear map $T : \Bbb R^m \to \Bbb R^n$ is uniformly continuous. My Try: We know that any linear map can be replaced by a matrix. Let $T(x) = Ax$ where, $A$ is the matrix for the ...
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3answers
29 views

Are the Unit Ball and Any other Ball Topologically Equivalent

How would I correctly show that the unit ball $B(0,1)\subset \mathbb{R}^n$ and the ball $B(a,r) \subset \mathbb{R}^n$ are Topologically Equivalent? I know I need to find a one-to-one function $f: ...
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32 views

Let $\{a_n\}$ be a sequence suppose there exists $0 <\lambda < 1$

Let $\{a_n\}$ be a sequence. Suppose there exists $0 < \lambda < 1$ such that $|a_{n+1} - a_n| \leq \lambda|a_n - a_{n-1}|$ for all $n >1$. Prove that $\{a_n\}$ converges. I seem to be ...
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31 views

Let $A$ be a subset of the metric space $X$, define $d(x, A) = \inf \{d(x,p)\; | \; p \in A\}$

If $A$ is a subset of the metric space $X$. For any point $x \in X$ define $d(x,A) = \inf \{d(x,p) \; | \; p \in A\}$. We have to prove a) If $x$ is an element of $A$ then $d(x,A)=0$. b) If $x$ is ...
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Proving that $\{u_k\}_{k=1}^\infty$, $u_k=\left\{1,\frac{1}{2},\frac{1}{3},\dots,\frac{1}{k},0,0,\dots\right\}$, does not converge in a metric space

Let $\ell$ be the set of sequences of real numbers where only a finite number of terms are different from zero $$\ell = \big\{\{x_n\}_{n=1}^\infty :x_i=0\text{ for all but a finite number of ...
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Compact sets and Open sets in a metric space

I have from reading up on things understood that open sets in a metric space is not compact. Though I have no clue why. I would like to know why is it they are not compact? I know that a compact set ...
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Homeomorphism from $(-1,1)$ to $\mathbb R$

I know that $f: (-1,1) \to \mathbb R$ defined by $f(x)=\tan \Big(\dfrac{\pi}2x \Big)$ is a homeomorphism . I am looking for some other homeomorphism between $(-1,1)$ and $\mathbb R$ which is not in ...
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Seeking to prove Continuity of $f(x) =\frac{x}{1+||x||}$

How would I prove that $f:\mathbb{R}^n\rightarrow B(\theta,1)$, where $f(x)=\frac{x}{1+||x||}$, is continuous? For metric spaces, I understand that if $f(x)$ is continuous at a point $p$ ...
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$(\Bbb N, d)$ and $(\Bbb N, \delta)$ are homeomorphic.

Let $\Bbb N \subset \Bbb R$ be given the induced euclidean metric $d$ and we consider $\Bbb N$ with the discrete metric $\delta$. To show: $(\Bbb N, d)$ and $(\Bbb N, \delta)$ are homeomorphic. I ...
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Open balls in euclidean space are homeomorphic to the whole space

The following question is from Fred H. Croom's book "Principles of Topology" Prove that each open ball $B(a,r), a\in \mathbb{R}^n, r>0$, considered as a subspace of $\mathbb{R}^n$, is ...
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$U \setminus V \neq \emptyset \iff \text{int}(U\setminus V)\neq \emptyset$ where $U$ and $V$ are open sets.

In order to solve a problem I need to prove that if U and V are open sets in a metric space then $U \setminus V \neq \emptyset \iff \text{int}(U\setminus V)\neq \emptyset$, but I'm not sure if it is ...
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1answer
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Equivalence and completeness of some metrics

Let $(X,d)$ be a complete metric space and $U$ be an open subset , $A:=X \setminus U$ , define a metric on $U$ as $$D(x,y)=d(x,y)+ \left|\frac1{\operatorname{dist}(x,A)}-\frac ...
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If any family of non-empty disjoint open sets is countable then the metric space is separable ? [duplicate]

I know that if a metric space $(X,d)$ is separable , then any family of non-empty disjoint open sets of the metric space is countable . Is the converse true ? That is if in a metric space , any family ...
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Why is a metric space an open subset of itself?

I've been reading about topology, and I've come across the following: Given a metric space $X$, the entire space $X$ is an open subset of $X$. I'm having some trouble thinking about this. I have a ...
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+50

Is It Always Possible to Draw A Connected Compact Set in $\mathbb R^2$?

Inspired by this answer, I wondered whether a printer could render all continuous functions "well enough". In particular, I am curious about the following statement: Let $S$ be a compact, ...
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2answers
25 views

Formula to best fit a rectangle inside another by scaling

I am very week in Math. I am a web programmer, and usually my work does not involve too much math - its more of putting records into database, pulling out reports, making those fancy web pages etc ...
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1answer
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Example of equivalent metrics on the same set such that uniform continuity of some function is not preserved

Give example of a set $X$ and two metrics $d_1,d_2$ on $X$ such that $(X,d_1)$ and $(X,d_2)$ are topologically equivalent but there exist a function $f:X \to X$ which is uniformly $d_1$ continuous but ...
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Can you build metric space theory without the real numbers?

Every metric space $M$ has a completion $\widehat M$, that is, it can be embedded as a dense subset in a complete metric space. When I first came across this theorem, I thought "well, that's amazing, ...
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3answers
50 views

Manhattan distance vs Euclidean distance

Suppose that for two vectors A and B, we know that their Euclidean distance is less than d. What can I say about their Manhattan distance?
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Measuring dispersion

I am trying to define a proper metric for characterizing dispersion of a set of $k \in \mathbb N$ points distributed over different spatial grids. Formally, given different 2-dimensional grids ...
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Need a help to show the following fuction is uniformly continuous.

$\newcommand{\dist}{\operatorname{dist}}$Suppose $ (X,\rho ) $ is a metric space and $ S $ is a non empty subset of $ X $. Then how to show the function $ g_S:X\rightarrow \mathbb{R} $ given by $ ...
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Showing $F(f)=\sin(f)$ is Continuous

We have a Banach Space $C[0,1]$; consider a function $F:C[0,1]\to C[0,1]$, where $$(F(f))(t):=\sin(f(t))$$ and this is $\forall t\in [0,1]$ Prove F is continuous. I tried showing that F was a ...
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1answer
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For $E \subseteq \mathbb{R}^n$, if every real valued continuous function on $E$ is uniformly continuous, then $E$ is closed and bounded.

Let $E$ be a subset of Euclidean space $\mathbb{R}^n$. Assume every continuous real-valued function of $E$ is uniformly continuous. Prove that $E$ is closed and bounded. The preceding exercise ...
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Show that there exists a real valued continuous function $f$ on $X$ such that $f(a) = \alpha$ and $f(b) = \beta $.

Let $a,b$ be two distinct points of a metric $(X,d)$ and $\alpha , \beta$ be any two given real number . Show that there exists a real valued continuous function $f$ on $X$ such that $f(a) = \alpha$ ...
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33 views

Uniform continuity of a continuous function

Let $(X,d);(Y, \rho)$ be metric spaces , $(X,d)$ have nearest point property and $f:X \to Y$ is a continuous function ; then is it true that $f$ is uniformly continuous on any bounded set $A$ in $X$ ...
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When is a bounded set in a metric space contained in a compact set?

If $A$ is a bounded subset of a metric space $(X,d)$ with nearest point property , then is it true that $A$ is contained in some compact set ? If $A$ is a totally bounded set of a metric space $(X,d)$ ...
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4answers
76 views

Showing that a finite or countable set in $\mathbb{R}^k$ is not connected

I have been using this result and I've looked in several books that all state this result but don't give a proof: Any finite or countable set in $\mathbb{R}^k$ is not connected. Can anyone ...
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1answer
55 views

Topologists Sine Curve problem

Topologists Sine Curve: $A:=\{(x,\sin \frac{\pi}{x}):0<x\leq 1\}\cup B:=\{(0,y):-1\leq y\leq 1\}$ I can show that it is connected.Problem is I cant show that it is path connected. Let $\gamma : ...
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2answers
57 views

Metric Spaces: Continuous, Unbounded Functions

The following question is from Fred H. Croom's book "Principles of Topology" Let $f:\mathbb{R}\rightarrow\mathbb{R}$ be a continuous, unbounded function. Show that there is a number $t_0$ for ...
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0answers
12 views

Adjacent angle Theorem in planes of constant curvature - easy geometric proof?

I am trying to proof the Adjacent angle Theorem in planes of constant curvature (2-Sphere, euclidean plane, hyperbolic plane) i.e given 4 points $a,b,c,d$ such that $d$ is lying on a shortest curve ...
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1answer
88 views

Can you write $R^2$ as a disjoint union of two totally disconnected sets?

Can you write $R^2$ as a disjoint union of two totally disconnected sets?
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23 views

Generalisations of properties of limits to metric spaces

I know the calculus lemma stating that If $\lim_{x\to x_0}f(x)=z_0$ and $g$ is continuous in $z_0\in\mathbb{R}$, then $\lim_{x\to x_0}g(f(x))=\lim_{z\to z_0}g(z)=g(z_0)$. It seems to me quite ...
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35 views

Why would I use Minkowski distance with p not 1, 2 or $\infty$

Minkowski distance is defined as: $$\left(\sum_{i=1}^n |x_i-y_i|^p\right)^{1/p}$$ where $p \geq 1$. I can understand that it can be useful for $p = 1, 2, \infty $, but in what practical applications ...