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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Borel sets: alternative characterization for metric space

For any topological space $(X,\tau)$, the Borel $\sigma$-algebra $\mathcal{B}$ is the $\sigma$-algebra generated by the open sets. In other words, it is the intersection of all $\sigma$-algebras on ...
2
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1answer
43 views

Understanding Rudin's proof that compact subsets of metric spaces are closed.

Rudin's Principles of Mathematical Analysis has the following definition of compact: A subset $K$ of a metric space $X$ is said to be compact if every open cover of $K$ contains a finite subcover. ...
5
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1answer
35 views

Show that the image of Lipschitz function $\gamma : [0,1] \to R^n$ has measure $0$, if $n \ge 2$.

Problem Statement: Let $\Gamma$ be the image of a Lipschitz continuous function $\gamma : [0,1] \to R^n$, that is, $\Gamma = \{\gamma(t) : t \in [0,1]\}$, and $|\gamma(t_1) - \gamma(t_2)| \le K |t_1 - ...
9
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7answers
480 views

What does it REALLY mean for a metric space to be compact? [duplicate]

I've been trying to wrap my head around the concept of compactness and get an intuitive understand of what it is. The definition used in my text book is the finite subcover definition. A subset ...
2
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1answer
42 views

Is the extended real line a metric space?

I've got a question reading the demonstration of the Theorem 3.2 in POMA of Rudin. Indeed, he says that every convergent sequence in a metric space is bounded. My question is: Is ...
0
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1answer
13 views

What can we do on $S$ in order that $H(S)$ be compact?

Let be $S$ a metric space. We define the hyperspace $H(S)$ as the metric spaces consisting of every no empty compact subset of $S$ and the Hausdorff metric. I want that $H(S)$ be compact imposing ...
1
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1answer
24 views

Distance between two ordered sets

Is there a way to measure the "distance" between two ordered sets? Say i got two sequences of letters: $$ S_1 \{A, B, C, D, E, F\} $$ $$ S_2 \{B, C, D, A, F, E\} $$ How could I find an "amount of ...
0
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0answers
21 views

Completeness of metric, normed and inner product spaces

For a metric space to be complete, it needs to have all cauchy sequences converge in the metric. 1) For a normed space to be complete, does it need Cauchy sequences to converge in the norm or in the ...
2
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1answer
11 views

Determining shapes in metric spaces?

I have a specific and a general question. My specific question is this: how would I determine the shape and location of the set of points satisfying $d(x,a) \leq 1$ in the metric space $(\mathbb{R}^2, ...
0
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0answers
29 views

munkres topology the meaning of the uniform metric on $\mathbb{R}^X$

I've been going through Munkres' Topology on my own, and I've come across an exercise where I can't even understand the question, it's in section 21, number 7. Let $X$ be a set, and let $f_n: ...
2
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1answer
30 views

Showing that $\mathbb{R}$ is a metric space with the metric $d(x,y)=|x-y|$

I want to show that $\mathbb{R}$ is a metric space with the metric $d(x,y)=|x-y|$. So three properties of the metric space $d(x,y)$ in general needs to be satisfied. My work: Let $x,y \in ...
2
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0answers
32 views

Definition of a length metric.

Let $(X, d)$ be a metric space. Define the induced intrinsic metric $\widehat{d}$ as follows \begin{equation} \widehat{d}(x,y) = \inf_{\gamma} \sup_{t_0, \ldots, t_N} \sum_{i=1}^{N}d(\gamma(t_{i-1}, ...
3
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2answers
89 views

Why is the metric $d(f,g)=\int_a^b|f(x)-g(x)|dx$ important?

The metric $d(f,g)=\int_a^b|f(x)-g(x)|dx$ appeared twice when I was studying. The author said that the space of Riemann integrable function with the metric $d$ is not complete, but the space $L^1$ ...
0
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0answers
25 views

elementary problem on uniform continuity in metric space [on hold]

Let $f \colon \mathbb{R}^n \to \mathbb{R}$ be the function defined by $f(x_1,x_2,\ldots,x_n)=\max\{|x_1|,|x_2|,\ldots,|x_n|\}$. Show that $f$ is uniformly continuous.how can i prove its a lipschitz ...
3
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1answer
35 views

If $X$ is compact and $C$ is an open finite cover of $X$ then $C$ has a maximum Lebesgue number.

Prove the following statement. If $X$ is compact and $C = \{U_\alpha : \alpha \in A\}$ is an open finite cover of $X$ then $C$ has a maximum Lebesgue number. Is my proof correct? Proof: Let $E$ be ...
0
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0answers
12 views

bijective uniformly continuous function from a subset of the Cantor set $K$ onto $X.$

Suppose $(X, d)$ is a non-empty metric space. Then $X$ is totally bounded if, and only if, there exists a bijective uniformly continuous function from a subset of the Cantor set $K$ onto $X.$ Proof: ...
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0answers
28 views

Topology of Metric spaces. [on hold]

I want to be clear about the concept of interior points, boundary points,limit points of a set in metric space. So, I want an explanation with examples on metric spaces. Thank You.
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2answers
68 views

Generalizing limits of sums, products, and quotients of sequences to abstract topological spaces?

Introductory real analysis books usually state some properties about limits of sequences. Suppose $\lim x_n=x$ and $\lim y_n=y$. Then: $\lim (x_n+y_n) = x+y$ $\lim c x_n = cx$, for $c \in ...
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0answers
22 views

Strongly Equivalent metrics [on hold]

How to show any two metrics to be strongly equivalent? Please suggest me the proper way to show this. Also i want to know how to find the constants in the respective definition.
1
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0answers
48 views

Determining the interior of $([-1, 1]\times[-1, 1])\setminus \{ y \in \mathbb{R}^2 : d((0, 0), y) < 0.25 \} \subseteq \mathbb{R}^2$

Let $M = (\mathbb{R}^2, d_e)$ be the metric space, with $d_e$ the Euclidean metric. Let $C \subseteq \mathbb{R}^2$ be defined by $$C = ([-1, 1]\times[-1, 1]) \setminus \{ y \in \mathbb{R}^2 : d((0, ...
3
votes
2answers
76 views

10 points outside a unit circle

Let $P_1$, $P_2$,... $P_{10}$ be ten points outside the unit circle centered at the origin $O$. Given that $\|P_iP_j\|\ge 1/\sqrt{2}$ for all $1\le 1<j\le 10$, find the minimum of the sum of the ...
2
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1answer
38 views

Prove that $1 / \min \{n\in\Bbb N :x_n\ne y_n \}$ is a metric on the set of all sequences of real numbers

Consider the set of all sequences of real numbers.For $x={(x_n)_n}$ and $y={(y_n)_n}$ we define $N(x,y)=\inf \{n\in\Bbb N :x_n\ne y_n,\text{if $x\ne y$} \}$. Now, $$d(x,y)= \begin{cases} 0, ...
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2answers
14 views

Bounded sequence in a metric space

I have a small question when we have a bouded sequence in a metric space; we say that there exists a closed ball $B'$ such that $(x_n)\subset B'$ or just there exist a ball $B$ such that $(x_n)\subset ...
3
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0answers
52 views

$f$ is continuous, $f : X \to X$, $X$ compact, and $f$ has an $\epsilon$-fixed point for each $\epsilon > 0$. Show $f$ has a fixed point.

Problem: Let $f : X \to X$ be a map from a metric space to itself. A point $z \in X$ is a fixed point of $f$ if $f(z) = z$. Let $\epsilon > 0$. A point $w \in X$ is an $\epsilon$-fixed point of $f$ ...
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votes
1answer
21 views

If $X$ is a polish space, how do we find an equivalent metric under wich $X$ is a totally bounded?

According to Stroock and Varadhan, If $X$ is a polish space, then one can choose an equivalent metric under which the space is totally bounded (see Stroock and Varadhan - Multidimensional diffusion ...
0
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0answers
41 views

Show that $y_n=x_{\phi(n)}$, defines a Cauchy sequence. [closed]

Let $\phi:\mathbb{N}\to\mathbb{N}$, such that $\displaystyle\lim_{n\to\infty}{\phi(n)}=\infty$. If $(x_n)$, is a Cauchy sequence in the metric space $M$, then $y_n=x_{\phi(n)}$, defines a Cauchy ...
0
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1answer
52 views

Example of Heine-Borel Theorem

For a subset $S$ of Euclidean space $\mathbb R^n$, S is closed and bounded if and only if S is compact (that is, every open cover of $S$ has a finite subcover). I need an example or ...
2
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1answer
46 views

subspace of a metric space

Let $(S,d)$ be a metric space, $\mathcal{S}$ the induced topology. $A\subset S$ a subset. It is easy to see that $A\cap\mathcal{S}=\mathcal{A}$, i.e., the topological subspace on $A$ is the ...
0
votes
1answer
36 views

How to determinate whether superset will be open or closed?

Let $M = (X, d)$ and A is closed subset of X, i.e. $A \subseteq X$. $A$ is told to be closed, iff it's complement $X\setminus A$ is open in $M$. But how can we determine, whether superset is open or ...
2
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3answers
51 views

Proving that a set is open using epsilons.

I'm trying to prove that the set $$A=\{x=(x_{1},x_{2})\in\mathbb{R}^2:x_{1}^{2}+x_{2}^{2}>1\}$$ is open in $\mathbb{R}^2$ with the usual norm is open with the definition of "epsilons". My attempt ...
0
votes
1answer
22 views

Find a surjective closed mapping that not an open mapping and a surjective open mapping that is not a closed mapping

Find a surjective closed mapping that not an open mapping and a surjective open mapping that is not a closed mapping Attempt: Suppose $X$ and $Y$ are metric spaces and $f : X \rightarrow Y$. We call ...
0
votes
2answers
54 views

Show that the mapping $f → f~'$ from $C^1([0 , 1])$ to $C([0 , 1])$ is not continuous.

Let $C^1([0 , 1])$ be the subspace of $C([0 , 1])$ consisting of the functions that have a continuous derivative throughout $[0 , 1]$. Show that the mapping $\Psi:f → f~'$ from $C^1([0 , 1])$ to $C([0 ...
4
votes
1answer
43 views

What has “$\delta$ Gromov hyperbolic space” got to do with $\mathbb{H}_n$?

The start of section $2$ of this paper, http://homes.cs.washington.edu/~jrl/papers/kl06-neg.pdf defines something called a ``$\delta$ Gromov hyperbolic space". Can someone explain what has this at ...
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votes
0answers
49 views

about cauchy sequence in metric space [closed]

Let $f$ be a function from a metric space $(X,d_1)$ to a metric space $(Y,d_2)$. If the image of every Cauchy sequence in $X$ is a Cauchy sequence in $Y$, how can I prove that $f$ is continuous?
4
votes
0answers
65 views

A discontinuous function $f: X \rightarrow Y$ satisfying: for each closed ball $B$ of $Y, f^{-1}(B)$ is closed in $X$

Find a function $f: X \rightarrow Y$ between metric spaces $X$ and $Y$ that is not continuous but has the property that for each closed ball $B$ of $Y, f^{-1}(B)$ is closed in $X$ Solution Attempt: ...
1
vote
1answer
28 views

What can we say about open unit balls of sup-norm and integral-norm

Consider the normed linear spaces $X_1=(C[0,1], ||.||_1)$ and $X_{\infty}=(C[0,1],||.||_{\infty})$ , where $C[0,1]$ denotes the vector space of all continuous real valued functions on $[0,1]$ and ...
0
votes
0answers
27 views

Banach contraction mapping to find unique solution of $y(x) = \int_0^1 G(x,s) f(s,y(s)) ds$

Problem statement: Use the Banach contraction mapping theorem to show that $$y(x) = \int_0^1 G(x,s)\ f(s,y(s))\ ds$$ has a unique, continuous solution $y = y(x)$ if (1) $G(x,s) \ge 0$ for $0 \le x, ...
2
votes
2answers
116 views

What is the “topology induced by a metric”?

My book gives the following definition: Let $(M,d)$ be a metric space, and let $\mathcal{T}$ be the collection of all subsets of $M$ that are open in the metric space sense... $\mathcal{T}$ is ...
0
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0answers
50 views

Hilbert Cube and Metric Space

Given that $d(x,y)=\sum_{n=1}^{\infty}2^{-n}|x_{n}-y_{n}|$ defines a metric on $H^{\infty}$ where $H^{\infty}$ is the Hilbert Cube, a collection of all real sequence $x=(x_{n})$ with $|x_{n}|\leq 1$ ...
2
votes
3answers
171 views

Are strongly equivalent metrics mutually complete?

Maybe I'm missing something, but I can't seem to find any references to my exact question. If two metrics, $d_1(x,y)$ and $d_2(x,y)$ are strongly equivalent, then there exists two positive constants, ...
2
votes
2answers
52 views

Show that the following mapping is a contraction.

I have the following problem from a past paper: "Show that the mapping, $$T(x_1,x_2)=\left(\frac{x_1+2x_2}5-1,\frac{x_1-2x_2}7+1\right)$$ is a contraction on $(\mathbb R^2,d_\infty)$." I ...
1
vote
2answers
68 views

How to determine whether those sets are open or closed?

Given those three sets below, A (left), B (center) and C (right), with A, B, C $\subseteq \mathbb{R^2}$, how can I determine, whether they are open or closed in metric space terminology via simplest ...
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votes
1answer
22 views

how to find the locus when distance from the origin is defined as d(x,y) = max { |x|,|y|},d(x,y) =a (where 'a ' is a non zero constant ) [closed]

How to find the locus when distance of any point from the origin is defined as d(x,y) = max {|x| |y|} where d(x,y) = a ( where is a non zero constant) I have a very long list of questions like these ...
0
votes
1answer
9 views

A point $a=(a_1,…,a_n)$ is isolated point in the cartesian product

Let be $M$ a metric space. A point $a=(a_1,...,a_n)$ is isolated point in the cartesian product $M=M_1\times...\times M_n$, if and only if, each coordinates $a_i$ is a isolated point in $M_i$ My ...
1
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1answer
17 views

Intersection of a dense set with an open set is dense in the open set

Let $A\subset M$ ann open subset, of the metric space M. If $X\subset M$ is dense in M, then $X\cap A$ is dense in A. My approach: If $X\subset M$, and $A\subset M$ is a open subset. Let be ...
2
votes
3answers
47 views

When the set of $r$-far interior points from a set is open

Let $E$ be a subset of a metric space $X$ and for $r > 0$ let $$ E_r = \lbrace x \in E : d(x,E^c) > r \rbrace .$$ Is the set $E_r$ always open? Equivalently, is the function $ x \mapsto ...
1
vote
1answer
33 views

Closed subspace of a metric space in which distance between any two points is at most $1$

$X$ be a metric space and $Y$ be a closed subspace of $X$ such that distance between any two points is at most $1$. Then $1$. $Y$ is compact $2$. Any continuous function from ...
0
votes
1answer
30 views

Prove complete metric space for $I=]0,\infty[$ with $d(x,y)=\lvert\ln(x)-\ln(y)\rvert$ [duplicate]

Let $I=]0,\infty[$ equipped with the metric $d(x,y)=\lvert\ln(x)-\ln(y)\rvert$, $\forall x,y \in I$. Prove that $(I,d)$ is complete. Any help, and thanks in advance.
-1
votes
1answer
33 views

Topology induced by discrete metrics and topology induced by singleton [closed]

Show that the topology generated by singleton sets is topology induced by discrete metric. $$d(x,y)= \begin{cases} 0,&\text{if } x=y\\ 1,&\text{if } x \ne y\\ \end{cases} $$
3
votes
2answers
76 views

Showing that a rectangle is equal to the closure of its interior

I'm trying to show that if Q is a rectangle, then Q equals the closure of Int Q. I have that the closure of Int Q is a subset of Q and I'm now working to show that Q is a subset of the closure of Int ...