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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Metric spaces with countable dense subset

Let $C^*(X)$ (endowed with sup norm) denote the metric space of all bounded real valued continuous functions on the metric space $X$. suppose that $C^*(X)$ contains a countable dense subset. I want to ...
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199 views

Zeroes of a continuous function on a metric space

Let $f$ be a continuous real valued function on a metric space $X$. Let $Z(f)$ be the set of all $p\in X$ such that $f(p)=0$ $\text{(a)}$ Prove that $Z(f)$ is closed. $\text{(b)}$ Recall ...
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Definition of accumulation point

I have here a definition of accumulation point: A point $x$ in a metric space $M$ is called an accumulation point of $A \subset M$ if every neighbourhood of $x$ contains some point of $A$ distinct ...
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91 views

Completion of a metric space

I got a doubt with the next exercise. Let $(X,d)$ be a metric space. Denote $\mathcal{B}(X,\mathbb{R})$ the subset of all bounded functions from $X$ into $\mathbb{R}$. Let $a \in X$. Show that ...
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127 views

How to define an interior point in terms of $\epsilon$-balls?

Which is the technically correct definition? I) An interior point of a set $B$ is a point that is the centre of some $\epsilon$-ball in $B$. II) An interior point of a set $B$ is a point that is in ...
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Precise definition of epsilon-ball

My textbook gives the following definition: "For each $\epsilon>0$, the $\epsilon$-ball about a point $x$ in a metric space $M$ is the set $\{y\in M:d(x,y)<\epsilon\}$." Is this correct? ...
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215 views

Properties for interior and closure in metric space.

I found the some following properties for general topology and prove these. But, I want to verify that the proofs are really true. Let $(X,d)$ be metric space. Let $A$ be any subset of $X$. Define ...
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1answer
196 views

Open dense subset of $\mathbb{R}$

Let $G$ be an open dense subset of $\mathbb{R}$ with the usual metric. Prove that for $x$ in $\mathbb{R}$ there exists $a$ and $b$ belonging to $G$ such that $x=a-b$.
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109 views

Show that the diagonal $\{(x,x): x\in X\}$ is closed in the metric space $(X\times X,d=\max\{d_X,d_X\})?$

Show that the diagonal $\{(x,x): x\in X\}$ is closed in the metric space $(X\times X,d=\max\{d_X,d_X\})?$ My attempt: Choose $(x,y)\in X\times X-\{(a,a): a\in X\}$ Then $c=d(x,y)/2>0.$ To ...
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53 views

Why do the author added the extra condition that $X$ needs to be $T_1?$

In my text it's written that, But I get to prove the result underlined red simply for a first countable space as: (N.B. by limit point the author wanted to mean the adherent point) ...
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150 views

Completing metric space

In the completion of a metric space, a distance is defined on the set of equivalence classes of Cauchy sequences: $$ \begin{align} \tilde d:\tilde X\times \tilde X &\to \mathbb{R^+}\\ ...
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1answer
344 views

Non-empty intersection of open balls in $R^n$ contain open balls

I want to prove that if the intersection of two open balls about the points $x, y$ (resp.) is non-empty, then there exists a third ball centered at some point $z\in B_{\epsilon 1}(x)\cap B_{\epsilon ...
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1answer
45 views

$X$ is a complete metric space, $Y$ is compact. $X \times Y$ is Baire?

Requesting a hint or solution. X is a complete metric space and Y is a compact hausdorff space. Trying to show that $X \times Y$ is a Baire space.
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1answer
109 views

Prove equivalence of definitions of “dense”

Prove that these two statements are materially equivalent (that is, one statement can be derived or proven from the other). Read below the statements for background information if you need it. Given ...
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219 views

Completion of a metric space in categorical terms

Is it possible to define the completion of a metric space using categorical terms?
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1answer
3k views

How to prove triangle inequality for $p$-norm?

Well, I've been studying metric spaces and to make the cartesian product of metric spaces a metric space I've heard of the $p$-norm defined in $\mathbb{R}^n$. So if $\mathcal{M}=\{M_i : i\in I_n\}$ is ...
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3answers
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How to show that the distance between these sets is positive?

Let $T_i=\{(1-t)x_i+ty_i;\;0\leq t\leq1\}$, where $x_i,y_i\in\mathbb{R}^n$; $i=1,2$. Could someone help me to prove that if $x_2= (1+\varepsilon)x_1$ and $y_2= (1+\varepsilon)y_1$ for some ...
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1answer
249 views

Clustering of Cofinally Cauchy nets

If $(X,d)$ is a metric space in which every Cofinally Cauchy sequence clusters. Does this imply every Cofinally Cauchy net clusters in the space?
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1answer
113 views

A result about connectedness and closed set.

Show that if $F$ is a closed and connected subset of a metric space $X$ then for every pair of points $a,b\in F$ and each $r>0$ there are points $z_0,z_1,\ldots,z_n$ in $F$ with $z_0=a$, $z_n=b$ ...
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155 views

Distance metric for (infinite) lines in 3D

I would like a metric $d(\;)$ between pairs of (infinite) lines in $\mathbb{R}^3$, with these properties: If two lines $L_1$ and $L_2$ are parallel and separated by distance $x$, then $d(L_1,L_2) = ...
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1answer
57 views

One Point Derivations on locally Lipschitz functions

Let $A$ be the algebra of $\mathbb{R}\to\mathbb{R}$ locally Lipschitz functions. What is the vector space of derivations at $0$? The proof that for continuous functions there aren't really any doesn't ...
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1answer
132 views

upper semi-continuity of a multi-valued function $T$ and lower semi-continuity of $d(x,T(x))$

Let $(X,d)$ be a complete metric space, $CB(X)$ the set of closed and bounded subsets of $X$, and $T:X\rightarrow C(X)$ be a multi-valued function. How can you prove this: If $T$ is upper ...
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1answer
101 views

How to describe the family $\tau$ of all open sets of $(\mathbb R^2,\delta)$

Ex. 1.2.65. Let $d$ be the Euclidean metric on $\Bbb R^2$. Define $$\delta(p,q):=\begin{cases} d(p,0)+d(q,0), & p\ne q \\ 0, & p = q, \end{cases}$$ for $p, q \in \Bbb R^2$. Show that ...
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geodesic metric

I'm trying to prove that the line segment is the minimizer of the distance $$d(x,y)=\inf l(\gamma),$$ where $x,y\in X$, $X$ is a Banach space, $\gamma$ is a path from $x$ to $y$ and ...
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Baby Rudin 2.26 Infinite subsets with limit points implies compactness

Having some trouble with this question. Let $X$ be a metric space in which every infinite subset has a limit point. Prove that $X$ is compact. Hint: By Exercises 23 and 24, $X$ has a ...
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119 views

Minkowski sum of two disks

An open disk with radius $r$ centered at $\mathbf{p}$ is $D(\mathbf{p}, r)=\{\mathbf{q} \mid d(\mathbf p, \mathbf q) < r\}$, and the Minkowski sum of two sets $A$ and $B$ is $A \oplus B=\{\mathbf p ...
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5answers
163 views

Find a metric space X and a subset K of X which is closed and bounded but not compact.

Find a metric space $X$ and a subset $K$ of $X$ which is closed and bounded but not compact. I can find a metric space $X$ like the below. Let $X$ be an infinite set. For $p,q\in X$, define ...
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Relationship metric space and $\sigma$-discrete base

Hy, I am newbie here. Can you help me to prove this proposition? If $X$ metric space, then there is a $\sigma$-discrete base $\mathcal{U}$ for the topology of $X$, i.e., ...
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1answer
54 views

I couldn't find the fault in $B_X(a,\epsilon)\times B_Y(b,\epsilon)=B_{X\times Y}((a,b),\epsilon)$

I know that the product of two balls of equal radius in metric spaces is not necessarily a ball in the product space. But I couldn't identify the fault in the proof where I showed ...
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A definition of metric space

Can you please help me solve the question below? I have no idea how to prove this one. Define the set $$X:=\{K\subset\mathbb C:K\text{ is bounded and closed}\}$$ Define a function $d\colon X \times X ...
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Metric Space (Elementary Analysis)

Let $d: X \times X \to \Bbb R$ is a function satisfying all properties of a metric space but $d(x,y)=0 \implies x = y$. If we define $\sim$ on $X$ by $x\sim y \iff d(x,y) = 0$, prove that $D([x], ...
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1answer
200 views

Lebesgue's criterion for Riemann integrability of bounded real valued functions defined on compact metric spaces

Let $(X, d)$ be a compact metric space, and let $S$ be the algebra of sets generated by the open and closed balls of $X$. Suppose we have a pre-measure defined on $S$ such that the measure of each ...
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1answer
118 views

I have to determine which of the following define a metric on $\Bbb R \,\,$?

I am stuck on the following problem: Determine which of the following define a metric on $\Bbb R$: $d(x,y)=\frac{|x-y|}{1+|x-y|}$ $d(x,y)=|x-2y|+|2y-x|$ $d(x,y)=|x^2-y^2|$ ...
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1answer
286 views

Examples of homeomorphisms between the real numbers and the positive real numbers?

I'm interested in homeomorphisms between the real numbers, $\mathbb{R}$, and the positive real numbers, $(0,\infty)$--where both spaces have the topology induced by the metric $d(x,y)=|x-y|$. Here ...
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2answers
180 views

Is there a structure theorem for nonempty, compact, nowhere dense subsets of the real line?

Let $X$ be the set of all nonempty compact nowhere dense subsets of the real line. Is there a theorem that describes the form of the elements of $X$? Context For open subsets of the line, such a ...
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221 views

why we want to use grassmannian space?

I wonder what's the special about grassmannian space? Why we want to use this space? On wikipedia, it says: "By giving a collection of subspaces of some vector space a topological structure, it is ...
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1answer
125 views

Prove triangle inequality

I want to prove that $d(x,y) = 1- \sum_i {\min(x_i, y_i)}$ where $\sum_i {x_i} = \sum_i {y_i} =1$ and $\forall i: x_i, y_i \geq 0$ satisfies the triangle inequality. The domain of $d$ therefore is ...
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1answer
108 views

Topologies coinciding at a point or a set.

Consider a set equiped with two topologies. What does it mean to say that the two topologies coincide at a point in the set? Is it meaningful to talk about this concept in general. Is it meaningful in ...
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Why is the discrete metric said to be so important

Can anyone enlighten me as to why the discrete metric is considered to be important in mathematics? The only real use I can see of it is that it shows the existence of a metric on any non-empty set. ...
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1answer
52 views

Extension of a metric defined on a closed subset

If $X$ is any metrizable space, $A$ is a closed subset of $X$. Let $d$ be a compatible metric on $A$ then $d$ can be extended to a compatible metric on $X$.
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When is a metric space Euclidean, without referring to $\mathbb R^n$?

Normally, the Euclidean space is introduced as $\mathbb R^n$. However, I've now been thinking about how one might define the $n$-dimensional Euklidean space only from the properties of the metric. ...
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Metrizability of a compact Hausdorff space

Show that a compact Hausdorff space is metrizable if the diagonal in $X\times X$ is a zero set of a non negative function ?
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Star graph embeddings

This is an homework question which I'm struggling with: Let $S = (V, E, w)$ a star graph, meaning, $S$ is a tree that all it's vertices are leafs except one. I need to : show that every weighted ...
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1answer
120 views

Cluster point of a sequence $\{x_n\}$ is the limit of some subsequence - Axiom of Choice? [duplicate]

In a metric space, a cluster point of a sequence $\{x_n\}$ is the limit of some subsequence. The only proof that I know works like this: Construct a sequence $\delta _k \to 0$. For each $\delta _k$ ...
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0answers
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Metric on the set of CDFs with finite p-th moment

Let $\mathcal{F}_p$, $p \ge 1$, be the set of all cumulative distribution functions of real valued random variables whose $p$-th moment is finite. I'm looking for a metric on $\mathcal{F}_p$ and ...
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1answer
183 views

a subclass of quasi metric spaces with properties almost identical to metric spaces

It is well known that passing from metric spaces to quasi-metric spaces (i.e., dropping the requirement that the metric function $d:X\times X\to \mathbb R$ satisfies $d(x,y)=d(y,x)$) carries with it ...
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Dual and completion of metric spaces

Say we have a metric space $(M,d)$, and we want to complete it in the following sense: Definition: A completion of $(M,d)$ is a complete metric space $(\widetilde{M},d')$ together with a Lipschitz ...
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212 views

Pre-compact balls of a separable metric space

Let $(X,d)$ be a complete metric space (I am not assuming that the metric is finite, there could be points in $X$ with infinite distance). Assume that each Ball in $X$ is pre-compact i.e. $\forall x ...
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239 views

Is the identiy function continous on equivalent metric spaces

Let $f$ be the identity function from $(X,d_1) \to (X,d_2)$. If $d_1$ and $d_2$ are equivalent metrics we can deduce that the identity function is continuous, right? Since for every open set $G$, ...
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$d$ is a metric on $X$ if $d(a,b) = 0 ⇔ a = b$ and $d(a, b) ≤ d(z, a) + d(z, b)$

The following is a question of Metric Spaces by O'Searcoid (pg 19) Suppose $X$ is a set and $d:X×X→\mathbb{R}$. Show that $d$ is a metric on $X$ if, and only if, for all $a,b,z ∈ X$, the two ...