0
votes
0answers
24 views

Point-set topology - derived sets, boundaries, etc.

Consider the metric space $(\mathbb R^2,\sigma)$, where $\sigma$ is the discrete metric. Let $$E=\lbrace(x,y)\in \mathbb R^2 : x^2+y^2<1 \rbrace$$ i.e. the unit disc. Find the following sets ...
-1
votes
0answers
16 views

Let Sn={$x\in Rn+1$; $\langle\ x,x\rangle$=1} a sphere n-dimensional. [on hold]

I study Metric Spaces and I have this problem. Let $\Bbb S^n=\{x\in \Bbb R^{n+1}; \langle x,x\rangle=1\}$ be a $n$-dimensional sphere. The projective space with dimension $n$ is the set $\Bbb P^n$, ...
1
vote
1answer
30 views

Where was I supposed to use compactness in this proof that a compact subspace can be separated from a point by open sets?

Can anyone check my proof below? P. Let $X$ be a metric space. Prove that if $K\subseteq X$ is compact and $x\notin K$, there exist disjoint open sets $U$ and $V$ such that $K\subseteq U$ and ...
1
vote
0answers
39 views

Point-set topology, characterizing sets

I think the question sort of speaks for itself. Please no solutions in the answers - I'm mostly looking to see if my logic makes sense: Let $E$ be a set in a metric space $(X,d)$. Let $E^{\circ}$ ...
0
votes
1answer
16 views

The measures used to define Hausdorf dimension versus Haar measure

I don't know if there is a name for the measure, so let me construct it: Given a metric space $X$ we define the outer measures $H_\delta^\alpha$ by $\forall A \subset X$ $H_\delta^\alpha ...
3
votes
2answers
45 views

Definition of neighborhood

I am starting to work through Rudin's Principles of Mathematical Analysis. For $(X, d)$ a metric space and $x \in X$, Rudin defines the neighborhood $N_r(x)$ of $x$ to be the set consisting of all ...
2
votes
2answers
55 views

Proof for distances to a set

With a metric space $(X,d)$, prove that $|d_E(x)-d_E(y)|\leq d(x,y)+d(y,z)$. In this context, $x \in X$, $d_E(x)=\inf\left\{d(x,z) : z \in E\right\}$, E is a subset of X. I've already proved the ...
0
votes
1answer
38 views

No direct proofs of “if $ f: (X, d_X) \to (Y, d_Y)$ is continuous and $X$ is compact then $f$ is uniformly continuous.”

I am studying the theorem "if $f:(X,d_X)\to (Y,d_Y)$ is continuous and $X$ is compact, then $f$ is uniformly continuous." I am not looking for a proof, but I have an argument against any attempt at a ...
3
votes
0answers
38 views

Showing equality of sets in $C[a,b]$

The exercise states: Let $a,b\in\mathbb{R}$, $a<b$ and let $(C[a,b],\Vert\cdot\Vert)$ denote the vector space of continuous real functions on $[a,b]$ endowed with the uniform norm. Let ...
0
votes
0answers
45 views

convergence of a sequence of cauchy

I would like to ask something about the convergence of a Cauchy sequence in a space $X$ metric. There will be a metric space $X$ such that If $(x_n)$ cauchy sequence in $X$ then $(x_n)$ is not ...
2
votes
3answers
61 views

The “Circle” is a Vector Space?

Consider the set of angles $C = [0, \ 2\pi)$ and, for all $x,y \in C$, define the $sum$ operation as the sum modulo $[0, \ 2\pi)$. The identity element of the addition is the angle $0$. The inverse ...
2
votes
2answers
52 views

Some special Metric on R

Apart from usual and discrete metric is there a metric on R which satisfy: d(x, y) = d(x+r , y+r) where x and y are any real no. and r is arbitary real no. Similarly is there a ...
3
votes
0answers
64 views

Condition on Equality of closure of open ball and closed ball, suppose I have a counterexample

Let $(X, d)$ be a metric space. Also for $x \in X$ and $r \ge 0$ define: $$ B(x,r) = \{ y \in X : d(x,y) < r \} \quad \mbox{ and } \quad K(x,r) = \{ y \in X : d(x,y) \le r \}. $$ Denote by ...
0
votes
1answer
71 views

Continuity, and continuity in topology.

Metric spaces: Neighborhood of a point $a$ is a Set of point $N$, such that $\exists\delta>0:B_\delta(a)\subset N$ ($B_r(x)$ = open ball at x of radius r) Definition of open set: "A subset $O$ of ...
0
votes
1answer
33 views

Definition of a metric-nonnegativity condition

There is a question in my mind which seems to be silly but I am desperately wanting the answer. Why a metric is defined from $X\times X$ to $\mathbb R$ and not to the set of nonnegative reals? I ...
1
vote
0answers
38 views

amazing boundedness problem from maximal function

Let $n\geq 2$. For any $M>1$, prove that there exists a constant $C_M>1$ such that for any ball $B$ in $\mathbb{R}^n$, if we denote $MB$ as the concentric ball of $B$ with $M$ times radius of ...
2
votes
0answers
31 views

Isomorphism isometries between finite subsets , implies isomorphism isometry between compact metric spaces

Let's $(X_1,d_1), (X_2,d_2)$ be compact metric spaces such that for every finite subset of $X_1$ like $A$ (respectively any finite subset of $X_2$ like $B$ ) there exists a finite subset of $X_2$ ...
4
votes
7answers
151 views

Convergence in a metric space

Is it possible to define a metric on $\mathbb R$ such that $(1,0,1,0,...)$ converges on $(\mathbb R, d)$? I believe it is impossible. But how to show analytically? Any hint would be appreciated.
0
votes
0answers
14 views

semi linear uniform space

In semi-linear uniform space, if $f$ is a function from $(X ,Γ_X)$ to $(Y,Γ_Y)$ that is linear and bounded ,is $f$ then continuous? Is the converse true?
2
votes
0answers
60 views

$ d(x,y)^2 \le d(x,z)^2 - d(y,z)^2\color{}{+\varphi\big(x,y,z,d(x,y),d(y,z) \big)}? $

In a metric space $(M,d)$ the triangle inequality $d (x, z) \le d(x, y) + d (y, z)$ gives us's the inequalitie $$ \quad d(x,y)^2 \ge d(x,z)^2 - d(y,z)^2\;\color{}{{-2\cdot d(x,y)\cdot d(y,z)}} $$ ...
0
votes
1answer
67 views

Prove that A is both open and closed. [closed]

Let $X = \{ z : | z | \leq 1 \} \cup \{ z : | z - 3 | < 1 \} $ be a subset of $\mathbb{C}$. Let the metric be the usual metric $d(x,y) = | x-y |$. Prove that the set A = $\{ z : | z | \leq 1 \}$ ...
0
votes
1answer
22 views

Is an open connected subset of Euclidean space a countable sum of open precompact connected subsets?

Let $U$ be an open subset in $\mathbb R^n$. Then there exists a sequence $(U_n)_{n=1}^\infty$ of open precompact subsets of $\mathbb R^n$ such that $U_n \subset cl U_{n+1} \subset U$ and ...
2
votes
1answer
56 views

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 ...
0
votes
2answers
72 views

Question on two metric spaces properties

Question: Let $X$ be a set and let $d_1$ and $d_2$ be two metrics on $X$. Assume that there exists a constant $C > 0$ such that $d_1(x, y) \le C\, d_2(x, y)\ \ \ \forall x, y \in X$. ...
2
votes
1answer
36 views

Existence of fixed point

I will copy the definition I am using just to make things clearer. Def. Let $(X,d)$ be a metric space and let $F:A(\subset X)\rightarrow X$. We say F is a contraction if there exists $\lambda$ where ...
1
vote
1answer
43 views

Metric space and closed sets (book misprint?)

I am not sure if there is a misprint in this corollary or if I am not getting the idea right. Corollary. Let $X$ be a metric space and let $A\subset X$. Then A is closed in $X$ iff: $$ ...
-1
votes
2answers
43 views

A Fundamental Property of Metric Spaces …

Let $(X,d)$ be a metric space and $A\subset X$ and also suppose that $G$ is open in $X$ prove the identity: $$ \overline {G\cap A}=\overline {G\cap \overline A} $$ Proposition: The intersection of ...
-1
votes
1answer
59 views

Some Property of Cantor set?

Draw a Cantor set $C$ on the circle and consider the set $A$ of all chords between points of C. Prove that $A$ is compact. Is $A$ convex? The proof of first part goes as follows: As we know ...
1
vote
1answer
76 views

Topology. Why is $T^{-1}$ continuous?

Today we did this proof, but we could not finish it and our prof said that the end would be easy, but I could not finish this proof. Let $X$ be a $T_3$ space with a countable basis $B$. Then we ...
-1
votes
1answer
27 views

Some Topological Properties of Starlike Sets!

A subset $E$ of $\mathbb R^n$ is starlike if it contains a point $p_0$ (called a center for $E$) such that for each $q\in E$, the segment between $p_0$ and $q$ lies in $E$. For more information please ...
3
votes
2answers
144 views

Compact metric connected space

If I have a compact metric space $X$ such that for all $a,b \in X$, there are points $a:=x_1,...x_n=:b$ such that $d(x_i,x_{i+1})< \varepsilon$, then this space is connected. Somehow, I don't see ...
1
vote
1answer
17 views

Continous function on compact interval - bounded

Let $K$ be a compact interval in $\mathbb{R}$. Then every continous function $\phi :K\rightarrow \mathbb{R}^d$ is automatically bounded. Is this a consequence of; the image of a compact is compact ? ...
0
votes
1answer
30 views

Hausdorff distance and union of sets

Let $X$ be a metric space; $A_1$, $A_2$, $B_1$, $B_2$ be non-empty subsets in $X$. Let $d(\cdot,\cdot)$ be the Hausdorff distance between sets in $X$. Then $$ d (A_1 \cup A_2 , B_1 \cup B_2) \leq \max ...
0
votes
2answers
29 views

Something not working out for me in the continuity definition

I'm studying analysis and I've ran into this proposition saying that a function from a metric space X to a metric space Y, is continuous if and only if for every open set O in Y, the inverse image of ...
0
votes
1answer
21 views

To show closedness of a subset in a metric spaces

Let $(X, d)$ be a metric space and $p\in X$, $\delta>0$ be fixed. Let $$A=\{q \in X : p \in X, d(p,q)>\delta\}$$ How to show that $A$ is closed? I tried to show that directly by taking $A$'s ...
0
votes
2answers
33 views

Lipschitz does not imply fixed points

I have the following problem in mind: Let us say we have a function $f:X\rightarrow X$ (X is a complete metric space) and it respects that if $x\neq y$ then : $d(f(x),f(y))<d(x,y)$. My trouble ...
1
vote
1answer
27 views

Verify why this is not a metric $d(x,y)=\|x-y\|_p$ ( $\|x\|_p=p^{-h}$ if $x=p^h\dfrac{m}{n}$).

$d(x,y)=\|x-y\|_p$ $p$ is prime and $\|x\|_p=p^{-h}$ if $x=p^h\dfrac{m}{n}$, where $m, n$ are coprimes with $p$. This is not a metric because if $x=y=p^k\dfrac{m}{n}$, then $x-y=0=p^0\dfrac0n$. ...
1
vote
0answers
32 views

Differentiation of Radon measures

Assume $\ (X,d)$ is a locally compact metric metric space and $\ \nu,\, \mu$ are Radon measures on $X$. Then, suppose that the following hypothesis hold: $\ w\in ...
0
votes
1answer
62 views

Isometry is not surjective

According to the definition I am using, an isometry is a mapping $f:X \rightarrow Y$ between two metric spaces $(X,d_{X})$ and $(Y,d_{Y})$: $$ d_{Y}(f(a),f(b)) = d_{X}(a,b) $$ for all $a,b \in X $ I ...
1
vote
2answers
91 views

Strange Property of Ultrametric Spaces and Metric Completion

The following property of ultrametric spaces seems quite strange: (No new values of the metric after completion) Let $x_1, x_2, \ldots$ be a sequence in $X$ converging to $x \in X$. Suppose $a \in ...
1
vote
1answer
55 views

Cauchy sequence of functions and uniform convergence

If $\Omega$ is a bounded domain, and on $C(\bar{\Omega})$ we use the uniform distance $$d(f,g)=\max_{\bar{\Omega}} |f-g|,$$ a Cauchy sequence of functions (w.r.t. the distance $d$) converge and the ...
-1
votes
1answer
50 views

Does there exist a metric under which $\mathbb{R}$ is incomplete? [duplicate]

Does there exist a metric under which $\mathbb{R}$ is incomplete?
1
vote
1answer
49 views

Poincaré inequality on metric spaces

In this book, page 91, example 4.18, it is said that the space $$Y=\{z\in\mathbb{C}:\ |z|=1,\ \arg z\in [-T,T]\}$$ where $3<T<\pi$ does not support a strong Poincare inequality (see page 84 for ...
0
votes
3answers
92 views

modern analysis: metric spaces and $\varepsilon$-neighborhoods

Prove or disprove that $d(f,g) = ({\int_0^1 |f(x)-g(x)|^{2}dx})^{1/2}$, on $C[0,1]$ is a metric. If so, describe the $\varepsilon$-neighborhood.
0
votes
1answer
32 views

Prove $g$ is continous on metric space.

For:$(X,d)$ is a metric space , $f:X\to X$ is a continous function and $g:X\to R, x\mapsto g(x)=d(f(x),x)$. Prove that $g$ is a continous function. Definition: $f$ is continous at $x_0$ if only if ...
0
votes
0answers
30 views

Convergence of difference of two cauchy sequence

Let $(X,d)$ be metric space, not complete, and $x_n , y_n$ be Cauchy Sequences in $X$. Then is $d(x_n,y_n)$ convergent? I know that $d(x_n,y_n) \leq d(x_n,x_m)+d(x_m,y_m) + d(y_m,y_n)$, so it is ...
0
votes
0answers
65 views

sufficient conditions that a function has a fixed point

Let be $(X,d)$ a complete metric space and $f:X\to X$ with $d(f(x),f(y))<d(x,y)$. I want to show that in general $f$ has no fixed point. But if $(X,d)$ is a compact space, indeed $f$ has a ...
3
votes
2answers
62 views

Is this a metric on R?

For $x,y \in \mathbb{R}$, define $d(x,y) = \sqrt{|x-y|}$. Is this a metric on $\mathbb{R}$? It's clear that $d(x,x) = 0$ and $d(x,y) = d(y,x)$ for all $x,y \in \mathbb{R}$. The triangle inequality ...
2
votes
2answers
33 views

What is this space with infinitely many different points with distance $1$ between any two different points?

I'm reading Mac Lane's: Mathematics, Form and Function: [...] There are also bizarre examples - such as "a space" with infinitely many different points, with distance $1$ between any two different ...
0
votes
0answers
39 views

Sequence Criterion of Continuity and Divergent sequences

A function $f : (X, d_X) \to (Y, d_Y)$ between metric spaces is continuous iff $$ \lim_{n \to \infty} f(x_n) = f(\lim_{n\to \infty} x_n) $$ for each convergent sequence $(x_n)$ in $(X, d_X)$. So if I ...