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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Topologizing Borel space so that certain functions become continuous

Let $X$ and $Y$ be compact metric spaces. Let $f:X \to Y$ be a Borel measurable map and suppose that $T:X \to X$ is a homeomorphism. Can one change the topology on $X$ such that $X$ is still a ...
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34 views

When can you drop an inequality term when you have more than two?

I am working on a problem: $d$ and $d'$ are metric equivalents on a set $X$, meaning there exist $n > 0, n' > 0$ such that for all $x, y \in X$, $d(x,y) \leq n \cdot d'(x,y)$, $d'(x,y) \leq ...
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Pointwise convergence imply uniform convergence

I am trying to find a condition under which a sequence of continuous functions on a metric space (or more generally in a topological space) which point wise converge to some function f should imply ...
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44 views

Understanding proof that if $c_1 d_1(x,y) \leq d_2 (x,y) \leq c_2 d_1 (x,y)$ then $d_1$ and $d_2$ are topologically equivalent metrics

Theorem. If there are strictly positive constants $c_1$ and $c_2$ such that $$c_1 d_1(x,y) \leq d_2 (x,y) \leq c_2 d_1 (x,y)$$ for all $x,y \in X$, then $d_1$ and $d_2$ are topologically ...
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Problem in standard proof of continuity when pre-image is open?

I have seen several proofs of the fact that a function $f$ from a metric space $X$ to a metric space $Y$ is continuous if every open set on $Y$ has an open inverse image on $X$. When proving the ...
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25 views

Question about a metric space proving.

Let us have an $(X,d)$ metric space. $U, V \subset X$ are disjoint, and $U\cup V = X$. Let $$D(x,y)=\left\{\begin{array}{lll}d(x,y)+1&\quad&\text{exactly one of $x$ and $y$ is in $U$}\\ ...
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How do I decide that a sequence is convergent in a metric space? Following example below.

Is the following sequence convergent in ?(Metric space is the normal, euclidean space)
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45 views

Is the following sentence a metric space?

$$X=\mathbb C^n\qquad d_p(x,y)=\left(\sum_{i-1}^n\left|x_i-y_i\right|^p\right)^{1/p}$$ $$x=(x_1,\ldots,x_n),y=(y_1,\ldots,y_n)\in\mathbb C^n$$ Is this a metric space, if $0 < p < 1$? I ...
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48 views

Finding any $\delta$ that $||(y,s)-(x,t)||<\delta$ implies $s<||y||$

I want to show that the set $$S=\{(x,t)\in\mathbb{R}^n\times\mathbb{R}\;|\;\;t<||x||\}$$ is an open set. Let $(x,t)\in S$, so we have $t<||x||$. So we have to show that ...
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18 views

Show that in a Normed linear Space $X,\overline {B(x,r)}=B[x,r]$

Show that in a Normed linear Space $X,\overline {B(x,r)}=B[x,r]$ where $\overline {B(x,r)}$ is closure of the set $\{y\in X:||y-x||<r\}$ and $B[x,r]=\{y\in X:||y-x||\leq r\}$ $\overline ...
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$D:=\{(x,y):x^2+y^2<1\}$ is complete?

How to conclude whether the set $D:=\{(x,y):x^2+y^2<1\}$ is complete ? I thought in the straight forward process of using a Cauchy sequence say $(x_n,y_n)$ then could not proceed further?
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1answer
20 views

Facing difficulty in finding a counterexample to prove that the set SL$(n, \Bbb R)$ is not bounded in M$(n, \Bbb R)$ for $n \geq 2$.

Facing difficulty in finding a counterexample to prove that the set SL$(n, \Bbb R)$ is not bounded in M$(n, \Bbb R)$ for $n \geq 2$. Here SL$(n, \Bbb R)$ is the set of all $n \times n$ matrices whose ...
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How to show that metrics generate the same topology?

Let $(X, d)$ be a metric space, let $c$ be a positive real number, and define a new metric $d'$ on $X$ by $d'(x,y) = c \cdot d(x,y)$. Prove that $d$ and $d'$ generate the same topology on $X$. ...
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Midpoints and strictly intrinsic metric

I'm studying the proof of the Theorem 2.4.16 (page 42) of this textbook (A Course in Metric Geometry by D. Burago, Y. Burago and S. Ivanov); I quote the statement: Theorem 2.4.16. Let $ (X,d) $ a ...
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To show that the only nonempty subset of $\Bbb R$ which is both open and closed in $\Bbb R$ is $\Bbb R$. [duplicate]

To show that the only nonempty subset of $\Bbb R$ which is both open and closed in $\Bbb R$ is $\Bbb R$. Let $A$ be a non empty subset of $\Bbb R$ which is both open and closed. Let $x \in A$. Then ...
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Two norms $||.||_1$ and $||.||_2$ on a vector space $V$ are equivalent.

Two norms $||.||_1$ and $||.||_2$ on a vector space $V$ are equivalent iff there exist positive constants $C_1,C_2$ such that $$C_1||.||_1 \leq ||.||_2 \leq C_2||.||_1$$ for all $x \in V$. I have ...
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61 views

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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1answer
47 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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1answer
45 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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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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24 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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30 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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Proving that the metric space $((0,\infty),d)$ is complete, with $d(x,y)=|\ln x-\ln y|$ [duplicate]

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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52 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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27 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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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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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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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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42 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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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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37 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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1answer
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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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34 views

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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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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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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265 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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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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91 views

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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110 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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30 views

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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3answers
56 views

Need a help to show $g_S(x)=\operatorname{dist}(x,S)$ 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 $ ...
4
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1answer
94 views

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 ...