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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Qualifier problem: Completeness of Metric Spaces

I am working on old qualifier problems as a review, and I came across this one: Suppose there exists a continuous surjection $f:X_1 \mapsto X_2$, where $(X_1,d_1),(X_2,d_2)$ are metric spaces, such ...
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Category of metric spaces versus category of non-empty spaces

Denote by $\mathbf{Met}$ the category of metric spaces with metric maps as morphisms. A function $(X,d)\xrightarrow{\ f\ }(X',d')$ is called metric if for every pair of points $x,y\in X$ we have ...
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Prove that $[0,1]$ is not isometric to $[0,2]$.

Prove that $[0,1]$ is not isometric to $[0,2]$. Suppose there is an isometry $f:[0,1]\to[0,2]$. Since f is continuous and surjective, the only values for $f(0)$ and $f(1)$ are $f(0)=0$ and ...
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Has anyone seen this space before? Does it have a name?

See the space below (the set taken as a subspace of the plane). It sort of looks like a comb, but with the wrap-around portion added, and the lower left corner removed. What would be a good name ...
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35 views

Characterization of closed sets in metric spaces

Let $X$ and be $Y$ be two metric spaces and $f:X\to Y$ a continuous function. We know that if $A$ is a closed set in $Y$ then $f^{-1}(A)=\{x\in X, \ \ f(x)\in A \}$ is a closed set in $X$. Now if we ...
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If image of closure belongs to closure of image, how to show preimage of interior belongs to interior of the preimage?

Here is exactly what I mean: Define a function $f:X\rightarrow Y$ from a metric space $X$ to another metric space $Y$. If any subset $A$ of $X$ satisfies $f(\bar A)\subset \overline {f(A)}$, then for ...
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Sufficient conditions for closed infinite pasting lemma

It's well known that the pasting lemma for infinitely many closed sets is false. It's reasonably easy to cook up examples such that for $X = \bigcup X_i$ with $X_i$ closed in $X$ such that $\left. ...
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Why does the additive subgroup of $\mathbb{R}$ generated by $1$ and $\sqrt{2}$ contain arbitrary small elements? [duplicate]

Let $G\subset \mathbb{R}$ be the additive subgroup of $(\mathbb{R},+)$ defined by $G=\mathbb{Z}+\sqrt{2}\mathbb{Z}$. I want to prove that for every $\epsilon>0$ there exists an element ...
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27 views

A proof (result) verification of a statement in metric-spaces

Let $A$ be a set , in a metric-space $(X,d)$ , having no limit points ; then I'm trying to prove that every convergent sequence $(x_n)$ in $A$ is ultimately constant . Please see the proof and tell ...
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51 views

Completeness of the set of convergent sequences

It's a problem from the book "Topology of Metric Spaces", written by Kumaresan: "Show that the set $\textbf{c}$ of convergent sequences in the Normed Linear Space of all bounded real sequences under ...
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Connected $G_\delta$ non-singleton, proper subsets in a connected complete metric space with more than one point

This is a question related to my last; I have still not solved it. Maybe this one is easier: Suppose $X$ is a connected complete metric space with more than one point. Must $X$ contain a ...
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52 views

How to solve for the extrinsic variables of a one variable scaled conformal metric to an equivalent metric?

Given the following metric equivalence \begin{align} e^{2w(x_2)} \left( dx_1^2+dx_2^2 \right) = dy_1^2+dy_2^2+dy_3^2 \end{align} is their a known solution for the extrinsic variables $y_1(x_1,x_2)$, ...
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60 views

Connected open proper subsets of a connected complete metric space

Suppose $X$ is a connected complete metric space with more than one point. Must $X$ contain a non-singleton non-empty connected proper open subset?
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31 views

Order of refinement of an open covering of $X$, a metric space

If every finite open covering of a metric space $X$ has a refinement of order $\leqslant n$, is it true that every open covering does too? We say that a covering has order $n$ if $n$ is the largest ...
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Surjection of norms

Let $V$ be an infinite dimensional $\mathbb{C}$ (or $\mathbb{R}$) vector space. Suppose there exists two norms on $V$ such that \begin{equation*} \| \cdot\|_1 \leq \| \cdot \|_2. \end{equation*} Is ...
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The number of non-degenerate proper subcontinua in a non-degenerate continuum

A continuum is any compact connected metric space. A continuum is non-degenerate if it is not a single point. My question is thus this: How many non-degenerate proper subcontinua must a ...
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57 views

Proving the metric attains a minimum on a compact subset

Let $(X,d)$ be a complete metric space. Suppose $B \subset X$ is compact. Prove that for every $a\in X$ the minimum $\min_{b\in B} d(a,b)$ exists. I'm pretty sure you can do this by just using the ...
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Connected $G_\delta$ sets in a connected completely metrizable space with more than one point.

Suppose $(X,\tau)$ is a connected completely metrizable space with more than one point. Let $\mathbb{G}$ be the set of all connected $G_\delta$ subsets of $X$. And let $\mathbb{O}$ be the class of ...
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56 views

In a metric space a compact set is closed

I want to show the following: Let $X$ be a metric space. Show that every compact subset $Y$ of $X$ is closed. The idea is to show that $X\setminus Y$ is open. So, for any $x \in X\setminus Y$, I ...
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Local geodesics in uniquely geodesic spaces

Suppose $Y$ is a proper, uniquely geodesic metric space. In such a space, is any local geodesic in fact a geodesic? Here the terms "geodesic" and "local geodesic" are taken in the metric sense: a ...
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1answer
59 views

Do connected complete metric spaces always contain a path?

Does every connected complete metric space with more than one point contain a non-trivial path? The pseudo-arc is an example of a connected metrizable space without a path.
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Connected complete metric spaces with more than one point.

Does every connected complete metric space with more than one point have infinitely many closed balls? And is any closed ball in a connected complete metric space connected?
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if $A, B$ are open in $\mathbb R$ then so is $A+B.$

I am trying to find out a counterexample to the problem: if $A, B$ are open in $\mathbb R$ then so is $A+B.$ But I could not find any such counterexample. Please help me.
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Shapes bounded only by lines

What is a term for the set of geometric shapes in the plane, that are bounded by one or more continuous closed curves? This set contains simply-connected polygons and circles but also polygons with ...
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25 views

Function vs. Polynomial Space

I've been reading up on spaces and was wondering if there was a difference between those two terms? Intuitively it would seem they are the same, but just so I don't dig myself into a hole, I was ...
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Notation problem with a set of tuples and a metric

The first question: Assume we have tuples $T_i = (x_i, \vec{c}_i)$ ($x_i$ is the name of the object which is characterized by $\vec{c}_i$ in a d-dimensional space) and define a set of them $TS = ...
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Lorentz transformation and Minkowski metric

For the exam I'm trying to solve some problems. Today I found this exercise and need some help: For the group S0(1,1) of the Lorentz transformation I have $\phi \in \mathbb{R}$ and $A_{\phi}: ...
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1answer
43 views

What is the difference between a Metric Space and a Pseudo-Metric Space?

I was wondering if anyone had information that would help me better understand the difference, so that I can work better on: Interesting Metrics I took a look at Metric assuming the value infinity ...
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33 views

Showing that the minimum distance between a closed and compact set is attained

I have two subsets of $\mathbb{R}^n$, given by $K$ and $F$, $K$ is compact and $F$ is closed. I'm trying to show that $\inf\{ d(x,y) : x \in K, y \in F \}$ is attained. My ideas so far: I know that ...
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Is there a meaning to convergence, limits and closedness in pseudo-metric spaces?

A. A sequence ($x_n$) in a metric space $M$ is said to converge to the limit $x \in M$ if the distance between $x_n$ and $x$ converges to 0 as $n$ goes to infinity. What happens when $M$ is a ...
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Representative elements in the symmetric difference metric

The symmetric difference is a natural way to quantify the distance between measurable sets: $$d(S,T)=measure([S\setminus T]\cup[T\setminus S])$$ This is a pseudo-metric because there may be ...
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Is the geometric mean of two metric spaces a metric space?

Suppose there are two metric spaces $d_1$ and $d_2$ over the set $X$. For $x,y \in X$, is $d_3(x,y) =\sqrt{d_1(x,y)d_2(x,y)}$ a metric space? I am having trouble with the triangle inequality. It is ...
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Prove that this is a metric

$d:\Bbb C \times \Bbb C \to \Bbb R$ Defined by $$d(z,w) := 2\frac{|z-w|}{\sqrt{(1+|z|^2)(1+|w|^2) }},$$ prove that $d$ is metric in $\Bbb C$. I had proved $d$ satisfies the two conditions to be ...
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Isometries from $\mathbb{R}$ to itself.

Prove that every isometry from $\mathbb{R}$ to itself is either $j_a$ or $i \circ j_a$. Here, $j_a$ is defined as $x\mapsto x+a$, and $i$ is defined by $x\mapsto -x$. Also, we're assuming the ...
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1answer
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The real numbers as a completion of the rationals

The real numbers are the completion (i.e. Cauchy sequences modulo equivalence) of the rational numbers. I want to define the real numbers this way, but without using uniform spaces. The problem is ...
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How to check whether this function is continuous or not..?

Let A and B be two disjoint closed sets of any Metric space X.I have to construct a continuous function such that $f(x):= 0$ if x belongs to $A$ $f(x) = 1$ if x belongs to $B$ My idea is to use the ...
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The completeness assumption in Prokhorov's theorem

Originally, I encountered this question on Terence Tao's blog, where the following exercise is presented: Exercise 23 (Implications and equivalences) Let $X_n, X$ be random variables taking values ...
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Proving compactness of the extended complex plane

Prove that $(\overline{\mathbb C}, \overline{d})$ with $\overline{d}(z,z')=d(\phi(z),\phi(z'))$, where $d$ denotes the euclidean distance in $\mathbb R^3$ and $\phi$ is the inverse of the ...
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Showing infimum of distance is attained

I have a continuous map $f: X \to X$ on a compact metric space and I am trying to show that $inf \{ d(x,f(x)) : x \in X \}$ is attained. My thoughts so far are to use sequential compactness to obtain ...
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1answer
50 views

Can I take Inverse Limits as Cauchy sequences literally?

I have been told to think of inverse limits as "Cauchy Completions" under some metric, for instance through the construction of the p-adic numbers. This got me thinking, though, and I wonder if the ...
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32 views

Interesting Metrics

To preface, this question might come off as a bit silly, but it would serve me well to understand the concepts, and to actually get a good answer for this. How can I design an ideal metric for ...
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Characterization of continuity by subsequences.

I have a difficulty trying to prove the following proposition. Any help would be greatly appreciated. $\textbf{Prop.}$ Let $(X,d_1)$ and $(Y,d_2)$ be two metric spaces. A function $f:X\rightarrow Y$ ...
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Verifying the triangle inequality for a metric for hyperbolic space

I read that the formula $d(x,y)=\mathrm{arccosh}(1+\frac{2||x-y||^{2}}{(1-||x||^{2})(1-||y||^{2})})$, where $x,y$ are in the open unit ball of $\mathbb{R}^{n}$ and $||\cdot||$ denotes Euclidean norm, ...
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Completeness in a Category of Metric Spaces.

Is there a way to describe completeness within a category of metric spaces? The point is that I'd like to have a description of compactness in metric spaces by something of the form totally bounded + ...
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1answer
57 views

Subset of infinite connected set

How to proove that infinite connected set has got proper infinite connected subset?
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proove that if scattered set is boundary

I need help: how to proove that if scattered set is boundary set, then his closure also is boundary? Sorry about my english. Ann
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How to prove that a metric space is compact if it is complete and totally bounded?

How to prove that a metric space is compact if it is complete and totally bounded? Wiki wrote that it is a generalisation of Heine–Borel theorem but I can't prove it.
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Which of the following sets are dense in $C[0,1]$

Which of the following sets are dense in $C[0,1]$ with respect to sup-norm topology? $1$. {$f$$\in$ $C[0,1]$ : $f$ is a polynomial } $2$. {$f$$\in$ $C[0,1]$ :$f(0)$=$0$} $3$. {$f$$\in$ $C[0,1]$ ...
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Is distance between two sets equal to that between their boundary?

I am not sure if the statement below is true. The statement is: Let $(M,d)$ be a connected metric space and $A, B$ be two nonempty subsets of $M.$ Assume the boundary $\partial A$ and $\partial B$ are ...
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A question about closed ball in metric space

Question: Let $(M,d)$ be a metric space and $\Omega$ be a bounded open subset of $M.$ For every positive real number $\epsilon,$ let $$\Omega_{\epsilon}:=\{x\in\Omega \mid ...