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.

learn more… | top users | synonyms (1)

1
vote
1answer
87 views

If there exist open, disjoint sets $A, B$ in $X, d$, then is $\bar A \cap B = A \cap \bar B = \emptyset$?

Clearly $A, B$ are separated in $X, d$ if $A, B$ are closed and disjoint since $(A \cup A') \cap B = \emptyset$ and vice versa. For disjoint open sets, I cannot come to a conclusion. I tried ...
2
votes
2answers
137 views

Proof that a limit point compact metric space is compact.

If $(X,d)$ is limit point compact, show it is compact. I have found multiple proofs of this that first show that limit point compact implies sequential compact, which in turn implies compactness. I ...
0
votes
1answer
52 views

How do I prove that $f(x) = x^2 : (0, 1/2) \to (0, 1/2)$ is not a contraction mapping?

How do I prove that $f(x) = x^2 : (0; 1/2) \to (0; 1/2)$ is not contraction mapping? I'd like to prove this in $\mathbb{R}$ with the Euclidean metric.
1
vote
1answer
101 views

What does finite $\epsilon$-net stand for?

$\mathbf{\text{Definition}\,\,4.3.6\,\,}$ Let $A$ be a subset of a metric space. We say that $A$ is totally bounded if for every $\varepsilon\gt0$, we can find a finite number of points $x_i,i\le ...
2
votes
2answers
153 views

Proving that the closed unit square in the plane is compact.

My thoughts on proving this statement is as follows: Suppose $G_a$ is an open cover of $Q= [0,1] \times [0,1]$. For each $x$ in $[0,1]$, there is some ball around $x$ with radius $r_x$ such that it ...
2
votes
3answers
177 views

Show that interval $(a, b)$ is not open in $\mathbb{R}^2$

I know that interval $(a, b)$ is open in $\mathbb{R}$. To show that interval $(a,b)$ is open in $\mathbb{R}$, I have done so: Let it be $x\in (a,b)$. Enough to find an open ball containing the point ...
0
votes
2answers
49 views

How to prove properties of the family of closed sets in a metric space

I know that is true: Let $(X, d)$ a metric space. The family $\mathcal {U}$ of all open subsets of $X$ has these properties: $1)$ $\phi, X\in \mathcal {U}$; $2)$ $U_1, U_2 \in \mathcal ...
-2
votes
2answers
397 views

if $f(x) = x$ if $x$ is rational $f(x) = x^2$ if $x$ is irrational Then f is continuous at 0 and 1. [closed]

Let $f$ be a function on the closed interval $[0, 1]$ defined by $f(x) = x$ if $x$ is rational $f(x) = x^2$ if $x$ is irrational Then $f$ is continuous at 0 and 1.
0
votes
3answers
39 views

How to tell $\overline {(a,b)}=[a,b]$, $\overline{\{\frac{1}{n}:n=1,2,3,\ldots}\}=\{\frac{1}{n}:n=1,2,3,\ldots\}\cup \{0\}$

Morning reading a book that deals with metric spaces noticed this fact: Tell that $$\overline {(a,b)}=[a,b],$$ $$\overline{\{\frac{1}{n}}\}=\{\frac{1}{n}\}\cup \{0\}.$$ I do not know much about ...
2
votes
2answers
96 views

Is $A=\{ (x,y) \in \mathbb{R}: x^2 + y^2 \le 1 \}$ compact and complete metric space

Is $A=\{ (x,y) \in \mathbb{R}^2: x^2 + y^2 \le 1 \}$ compact and complete metric space in $(\mathbb{R}^2,d_c)$ where$$ d_c:A\times A\to\mathbb R,(x,y) \mapsto \begin{cases} d_e(x,y) &\text{if ...
0
votes
1answer
132 views

If the distance between any two points is less than $1$, must $X$ be compact?

Let $X$ be a complete metric space such that the distance between any two points is less than $1$. Then is $X$ necessarily compact? Thanks in advance.
0
votes
1answer
36 views

let (X,d) be a metric space. d is discrete iff X∩X'=∅

Let (X,d) be a metric space. prove that: (X,d) is discrete if only if X∩X′=∅,X′ is the set of all limit points of X
0
votes
2answers
64 views

Nowhere dense subset

Let $(X,d)$ be a M.S. without any isolated points and $A$ be a subset of $X$ such that each point is an isolated point of it. Show that $A$ is Nowhere dense.
0
votes
5answers
83 views

$\mathbb{N}$ is closed in $\mathbb{R}$, but $\mathbb{Q}$ is not.

I know that the subset $F$ for a metric space $X$ we say is closed if its complement $X\backslash F $ is a open set, but do not know how to solve the following example $\mathbb{N}$ is closed in ...
1
vote
1answer
30 views

Density of the function class

Let $X$ be any set and let $[0,1]^X$ (the class of all functions $X\to[0,1]$) be endowed with the metric given by $\rho(f,g):=\sup_{x\in X}|f(x) - g(x)|$. Consider any class of functions $\mathscr ...
2
votes
2answers
156 views

Show that $\displaystyle\prod_{\Bbb{N}} \Bbb{R}$ with the box topology is Hausdorff but not metrizable.

Show that $\displaystyle\prod_{\Bbb{N}} \Bbb{R}$ with the box topology is Hausdorff but not metrizable. $\Bbb{R}$ must be Hausdorff. For $x_1, x_2 \in \Bbb{R}$ (where $x_1 \not= x_2$), if $d$ ...
1
vote
3answers
79 views

Is the set of extended real-valued numbers open or closed

If I assume that my topology is defined on the extended real-valued numbers, then $\mathbb{R}\cup\left\{-\infty,+\infty\right\}=\left[-\infty,+\infty\right]$, acting as my entire space, is both open ...
0
votes
0answers
32 views

There exist points of minimal distance between closed and bounded $A$ and closed $B$ [duplicate]

I need to prove the following: Let $A$ be a bounded and closed set, and let $B$ be a closed set. Show that there exist $a_0$ in $A$ and $b_0$ in $B$ such that - for each $a \in A$ and $b \in B$: ...
1
vote
1answer
215 views

Proving the existence of a non-monotone continuous function defined on $[0,1]$

Let $(I_n)_{n \in \mathbb N}$ be the sequence of intervals of $[0,1]$ with rational endpoints, and for every $n \in \mathbb N~$ let $E_n=\{f \in C[0,1] : f \:\text{is monotone in}\: I_n\}$. Prove that ...
1
vote
4answers
64 views

Prove B is a closed subset of X given the f and g are continous?

Let $(X;\rho)$, $(Y;\sigma)$ be metric spaces. Let $f,g : X \to Y$ be continuous. Prove that the set $B=\{x\in X: f(x)=g(x)\}$ is a closed subset of $X$
1
vote
1answer
45 views

Connected components of a given space

For every $n \in \mathbb N$, let $A_n=\{\frac{1}{n}\}\times[0,1]$, and let $X=\bigcup_{n \in \mathbb N} A_n \cup \{(0,0),(0,1)\}$. Prove that: i)$\{(0,0)\}$ and $\{(0,1)\}$ are connected components of ...
2
votes
1answer
279 views

looking for proof that this uniformly bounded sequence of functions has no pointwise convergent subsequence

Math people: I couldn't find a similar question, so here goes: I would like to prove the fact (?) that the sequence of functions $(f_n) \subset C([0,1])$ defined by $f_n(x)=\sin(nx)$ does not have a ...
2
votes
2answers
138 views

Prove that the identity map $(C[0,1],d_1) \rightarrow (C[0,1],d_\infty)$ is not continuous

$$d_\infty = \max|x_i - y_i|$$ $$d_1 = \sum_{i=1}^n |x_i - y_i|$$ The first part of this question was to prove that the identity map $$(C[0,1],d_\infty) \rightarrow (C[0,1],d_1)$$ is continuous, ...
0
votes
2answers
37 views

*Find examples in which $\mathbb{R^2}$ which infinite intersection of open sets is not open set*

Could please someone to help me $1.$ Find examples in which $\mathbb{R^2}$ which infinite intersection of open sets is not open set. $2.$ Find a set in $\mathbb{R}$ as $a)$ not open not closed; ...
0
votes
1answer
51 views

Real Analysis Closed and Bounded Set Question

Suppose K is a nonempty closed and bounded subset of a metric space X and x $\in$ X. Show the following hypothesis fails: There is a p $\in$ K such that, for all other q $\in$ K, d(p,x) $\leq$ d(q,x). ...
2
votes
2answers
318 views

Show that the topological space ( X, $\tau$ ) is not metrizable

For the topological space ( X, $\tau$ ), with X = {0, 1} and $\tau$ = { $\emptyset$ , {0}, {0,1} } , prove that ( X, $\tau$ ) is not metrizable. I know intuitively it can't be but don't know how to ...
0
votes
2answers
35 views

convexity and the interior sphere condition

Consider $\Omega $ a open, convex bounded subset of $R^n$. Let $x_0 \in \partial \Omega$. I believe that exists a open ball $B \subset \Omega$ such that $\partial B \cap \partial \Omega = \{ x_0 \}$. ...
4
votes
1answer
63 views

Free Metric Space?

Do free metric spaces exist? Ie.: An object in the category of metric spaces and lipschitzian maps. If so would these be the complete metric spaces, since they satisfy a similar universal property? ...
1
vote
1answer
37 views

Whether a function$d(m,n)=\left\vert\frac{1}{m}-\frac{1}{n}\right\vert$ metrics

I saw in a magazine the following example" Whether a function $d(m,n)=\left\vert\frac{1}{m}-\frac{1}{n}\right\vert,$ where $m,n\in\mathbb{N}$ metrics. I know that map $d:XxX\rightarrow\mathbb{R}$ ...
0
votes
1answer
239 views

Show that $C[a,b]$ is a complete space under the metric $d(f,g)=\sup_{t\in [a,b]}|f(t)-g(t)|$.

$C[a,b]$ is a normed vector space of all continuous complex valued functions on $[a,b]$, with supremum norm $$\|f\|_\infty=\sup_{t\in [a,b]}|f(t)|.$$ The metric induced by the norm is ...
0
votes
2answers
74 views

Open sets of sequences

Let $M$ denote the space of sequences $(x_n)$ where $x_n \in\{0,1\}$ for each $n$. Let $$d\colon M\times M\rightarrow\mathbb{R}\colon ((x_n),(y_n))\mapsto\sum_{i=1}^{\infty}|x_i-y_i|2^{-i}$$ be the ...
0
votes
1answer
493 views

Complete metric spaces - continuous functions

Let $(X,d_X)$ and $(Y,d_Y)$ be metric spaces with $(Y,d_Y)$ bounded. Let $C(X,Y)$ denote the set of all continuous functions from $X$ to $Y$. Let $d$ be the uniform metric on $C(X,Y)$, i.e. $d(f,g) = ...
5
votes
3answers
147 views

Extending a connected open set

Assume $\emptyset\neq V\subseteq U\subseteq\mathbb{R}^n$ are open and connected sets so that $U\setminus\overline{V}$ is connected as well. Given any point $x\in U$, is there always a connected open ...
0
votes
2answers
59 views

the power series converges in compact convergence topology

Consider the sequence of functions $f_{n}: (-1,1) \rightarrow R$ defined by:$$f_{n}(x) = \sum_{k=1}^{n}{kx^{k}}$$ a) Prove that $(f_{n})$ converges in the topology of compact convergence, ...
0
votes
1answer
34 views

$B(R,R)$ is not closed in the topology of compact convergence

I'm doing this exercise in Munkres book, and got no clue to solve this problem. Help someone can help me. Let $B(R,R)$ be the set of bounded functions $f: R \rightarrow R$. Prove that ...
0
votes
1answer
186 views

relative compact implies totally bounded?

Let $M$ be a metric space. It's always true that if $A$ is relative compact (i.e $\bar{A}$ is compact) then $A$ it's also totally bounded?. I tried to proved it, considering the finite subcovering of ...
2
votes
1answer
52 views

Some questions on convex sets.

Are all bounded closed convex sets in a metric space $(M,d)$ compact? or if not are they complete? The positive definite matrices form a convex set (Why does a positive definite matrix defines a ...
3
votes
1answer
91 views

Metric spaces and limit points question?

Let $X, d$ be a metric space. For each $x \in X$ and nonvoid $A, B \in X$, define $$d(x, A) = \inf\{d(x, a) : a \in A\}$$ and $$d(A, B) = \inf\{d(a, b) : a \in A, b \in B\}$$ Prove that $d(x, A) = 0$ ...
1
vote
1answer
46 views

Metric on an infinite dimensional space with equivalence relation.

In analyzing a problem I've come across a space defined by the following equivalence relation: $(\cdots, x_{-2}, x_{-1}, x_0, x_1, x_2, \cdots) \sim (\cdots, z^{-2}x_{-2}, z^{-1}x_{-1}, x_0, zx_1, ...
4
votes
0answers
91 views

Completeness of a metric space with the Hausdorff metric

Let $(Y,d)$ be a metric space and let $K(Y)$ denote the set of all non-empty compact subsets of $Y$. This collection is a metric space when equipped with the Hausdorff distance $h$. I want to prove ...
3
votes
1answer
70 views

What type of convex constraint is defined by SQRT?

Let $A$ be an $n \times n$ positive semidefinite matrix and $\forall k, x_k \in \mathbb{R}^n$. The distance with respect to this matrix is defined as $ \|x_i -x_j\|_A := ...
0
votes
2answers
152 views

Prove that open subspace of a topologically complete space is topologically complete

I'm trying to prove that an open subspace of a topologically complete space is topologically complete. I follow the hint in the book. We defined $\phi : U \rightarrow R$ by the equation $$\phi(x) ...
8
votes
2answers
388 views

A metric space such that all closed balls are compact is complete.

I am trying to solve the following exercise: Let $(X,d)$ be a metric space that has the property that for any $x\in X$ and $r>0$, the closed ball $$\bar{B}(x,r):=\{y\in X:d(x,y)\leq r\}$$ ...
0
votes
1answer
35 views

$\sup_n \inf_{y \in X} (F(y) + n d(x,y))= F(x)$

Let $(X,d)$ be a metric space and $F : X \rightarrow [0, +\infty)$ a lower semicontinuous function. Then $$ \sup_n \inf_{y \in X} (F(y) + n d(x,y))= F(x). $$ Is this true? Intuitively it works since ...
5
votes
3answers
159 views

Does $d(x_{n+1},x_n)<d(x_n,x_{n-1})$ imply anything?

Let $(X,d)$ be a complete metric space, $(x_n)_{n\in\mathbb{N}}\subset X$ such that $d(x_{n+1},x_n)<d(x_n,x_{n-1})$ for all $n\in\mathbb{N}$. Since I cannot construct such sequence which is not ...
1
vote
1answer
247 views

Continuous images of Cauchy sequences are not necessarily Cauchy

Could you please provide an example for two metric spaces $X,Y$, a continuous function $f$ that maps $X$ to $Y$ and a Cauchy sequence in $X$, which is not mapped to a Cauchy sequence in $Y$ by $f$? ...
6
votes
2answers
359 views

Every open ball is connected

Let $(X,d)$ be a metric space such that for all $x \in X$ and all $r>0$, $\overline{B(x,r)} = \{y \in X \mid d(x,y)\leqslant r\}$ Show that every open ball of $X$ is connected. Note- I was trying ...
1
vote
2answers
45 views

Showing that a set is open

Endow $R^2$ with the metric $d(a,b)$ ={ $max{|a_1-b_1|,|a_2-b_2|}$} where $a$=$(a_1,a_2)$ and $b$=$(b_1,b_2)$. Show that $S$={${a \in R^2|a_1^2+a_2^2<1}$} is open in $R^2$ with this metric. $S$ ...
-1
votes
1answer
48 views

Showing that a set is open/closed

$\def\R{\mathbb R}$ Is the set $$S=\{(x_1,x_2,x_3) \in \R^3 \mid e^{x_1} + x_2^2 <x_3 \} \subset \R^3$$ open or closed? My attempt: Let $f:\R^3 \to \R$, $f(x_1,x_2,x_3)$ =$e^x_1 + ...
0
votes
1answer
43 views

Showing that a set is closed

Show that the set $S=\{a \in \mathbb{R}^3\,| \,a_1 +a_3^2 \sin(a_1+a_2)\geqslant a_3\}$ in closed in $\mathbb{R}^3$ with the euclidean metric. I know that I would probably have to show that the ...