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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What do the eigenvalues/vectors of a metric describe?

Given a finite metric space $(X = \{ x_i \}_{i=1}^n,d)$, one can form the matrix $A$ of pairwise distances $a_{ij} = d(x_i, x_j)$. What does the eigenspectrum of this matrix say about the metric $d$? ...
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49 views

On the cardinality of $\mathbb R \times …\aleph_1 {times}$ and $\mathbb R \times …2^{\aleph_0} \space {times}$

I think I can prove that closure of every countable set in any metric space has cardinality at most $\mathcal c=2^{\aleph _0}$ . So if a metric space is separable i.e. has a countable dense subset $A$ ...
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A metric space of which the geodesic is not a metric

The text book in my course has an exercise about finding a metric space whose (usual) length metric is not a metric. It wants me to find a metric space $(X,d)$ satisfying $d'(x,y)=0 \ \ $for some ...
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Proving the usual distance metric in $\mathbb{R}$ is complete

If we allow the metric to be $d(x,y)=|x-y|$, we must prove that this is complete. Now, I have proven all properties of a metric space. However, I don't particularly now where to begin to prove that ...
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25 views

Confusion about notation for a metric space.

My professor wants me to prove that $(\mathbb{R},|\cdot|)$ is a complete metric. Now, I know how to do so, but I am confused as to what she is referring to by $$|\cdot|$$ Is she referring to the norm ...
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39 views

Are $\{(x,0) \in \mathbb{R^2} : x \in \mathbb{R}\}$ and $\{(x,\frac{1}{x}) : x >0\}$ separated?

Our definition of separation is: If $C$ is a subset of a metric space, then $(A, B)$ is a separation of $C$ if $C = A \cup B$, $A \neq \varnothing$, $B \neq \varnothing$, and we cannot have that ...
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22 views

Is it true that every 1st category subset of a 2nd category space has empty interior?

Let $X$ be a metric space. Are these conditions equivalent: Each set of the 1. category in $X$ has empty interior; $X$ is of the 2. category. It is obvious that $1 \Rightarrow 2$. Is it true that ...
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Is the following metric space is a complete metric space?

We have $X=\ell^1$, which contains sequences, which are absolutely convergent, and $d(a_n,b_n) = \sum_{k=1}^{\infty}|a_k-b_k|$. Is this metric space complete or not?
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26 views

Show the Euclidean metric and maximum metric are strongly equivalent.

I need to show that the Euclidean metric and maximum metric (or square metric??) are strongly equivalent. I have no idea how to start this proof. Any help? $d_1, d_2$ are called strongly equivalent ...
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32 views

Let $(X,d) ; (Y,e)$ be two metric spaces ; can we define a metric on $X \cup Y$ whose restriction on $X$ is $d$ and restriction on $Y$ is $e$ ?

Let $(X,d) ; (Y,e)$ be two metric spaces ; can we define a metric $\rho$ on $X \cup Y$ such that $\rho(x,y):=d(x,y) , \forall x,y \in X$ and $\rho(x,y):=e(x,y) , \forall x,y \in Y$ ?
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How to find the cardinality of the set of all isolated points of $X$?

Suppose $X$ is an uncountable complete metric space .Can the cardinality of the set of all isolated points of $X$ be countably infinite?
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26 views

If there exists an open set $U$ in $X$ such that $A = Y \bigcap U$ then $A $ is open in $Y$

Let $Y$ be subspace of a metric space $X$. Show that $A \subset Y$ is open in $Y$ if and only if there exists an open set $U$ in $X$ such that $A = Y \bigcap U$. My Try: Let $A$ be open in $Y$. Then ...
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29 views

Colorings of Topological Partitions (color-boundedness)

Let $(X,\tau)$ be a topological space. DEFINITIONS: Define a topological partition of $X$ into connected sets to be a collection of pairwise disjoint open connected sets $\{U_i\}_{i\in I}$ such that ...
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35 views

Colorings of Topological Partitions (Path adjacency)

Let $(X,\tau)$ be a topological space. DEFINITIONS: Define a topological partition of $X$ into connected sets to be a collection of pairwise disjoint open connected sets $\{U_i\}_{i\in I}$ such that ...
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$f:X\longrightarrow\mathbb{R}$ is continuous iff $\{x\in X:f(x)\geq\alpha\}$ and $\{x\in X:f(x)\leq\alpha\}$ are closed $\forall\alpha\in\mathbb{R}$

I'm trying to prove that $f:X\longrightarrow\mathbb{R}$ is continuous if, and only if, the sets $$\{x\in X:f(x)\geq\alpha\} \text{ and } \{x\in X:f(x)\leq\alpha\}$$ are closed ...
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A question about compact and complete metric spaces [duplicate]

I have a question about compact and complete metric space. These two concepts how related to each other. Is compact metric space complete? If the question is elementary I apologize you. Thank ...
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1answer
60 views

Neighborhoods and Metric Spaces in Real Analysis

In my analysis class, we are looking at metric spaces and topologies. We were asked to prove the following two theorems. I believe I have the first one completed, but am somewhat confused by the ...
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40 views

Doubt about a metric space analysis task

We have $X=\left\{(x,y)\in\mathbb R^2:x^2+y^2\le9\right\}$, with the standard $d$ euclidean metric. Let $B_2 = X$, and we have: $$B_1=B_2\cap\left\{(x,y)\in\mathbb R^2:(x-2)^2+y^2\le16\right\}$$ ...
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A set $U$ is open iff it is union of open balls

Let $(X,d)$ be a metric space. Consider the collection $\mathcal{T} = \{ U \subset X: \forall u \in U, \exists r>0 \; \; , B_r(u) \subset U \}$. We showed that $(X, \mathcal{T} )$ is a topological ...
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43 views

Prove that a function that maps a discrete metric space to any metric space is continuous [closed]

Let $f:D→M$ where $M$ can be any metric space and $D$ is any set with the discrete metric. Prove that $f$ is continuous. I'm not sure where to begin with this.
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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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46 views

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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1answer
29 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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1answer
32 views

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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24 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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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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16 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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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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77 views

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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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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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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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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23 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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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 $d(x,y)=|\ln x-\ln y|$

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
41 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
33 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
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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1answer
26 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 ...
2
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1answer
46 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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1answer
38 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 ...