4
votes
7answers
144 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
12 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
46 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 $$ \quad d(x,y)^2 \ge d(x,z)^2 - d(y,z)^2\;\color{}{{-2\cdot d(x,y)\cdot d(y,z)}} $$ ...
1
vote
1answer
63 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
19 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
44 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
71 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 $d_1(x, y) \le C\, d_2(x, y)\ \ \ \forall x, y \in X$. ...
3
votes
1answer
34 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
55 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
23 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
140 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
21 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
27 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
19 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
32 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
29 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 ...
1
vote
1answer
50 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
80 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
52 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
48 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
43 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
31 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
27 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
64 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
59 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
36 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 ...
0
votes
1answer
33 views

Show that $d_g$ is a metric on $l^1$.

On the space $l^1$ of complex valued sequences $(x_n)$ such that $\sum|x_n|<\infty$, define for $x=(x_1,x_2,\cdots)$, $y=(y_1,y_2,\cdots)$ the metric $d_f$ by ...
4
votes
2answers
111 views

Show that $d(x,y)$ in a metric on $X$.

Let $d_a(x,y)=7|x-y|$ and $d_b(x,y)=|x+y|$ be metrics on set $X$. Show that $d(x,y)=d_a(x,y) + d_b(x,y)$ is also a metric on $X$. Would I be correct in writing $d_a(x,y) + d_b(x,y)$ as $7|x-y| + ...
0
votes
0answers
35 views

Example of a sequence function which is unbounded

Let $(X,d)$ be a metric space and $T:X\to X$ be self map on $X$. A map $\phi :X\to[0,\infty)$ is said to be sequence function with respect to $T$ if $$\lim_{n\to\infty}\phi(x_n)<\infty$$ whenever ...
1
vote
1answer
18 views

Metric on half-open interval s.t. subset is open w.r.t. $d$ iff open w.r.t. Euclidean metric

I wish to find a metric $d$ on the space $X = (0,1]$ such that $(X,d)$ is complete and so that a subset of $X$ is open with respect to $d$ if and only if it is open with respect to the Euclidean ...
0
votes
2answers
27 views

Convergence of $f_{n}(x)=\frac{1}{x^2+n^2}$ and $g_{n}(x)=\frac{2nx}{x^2+n^2}$ in sup norm

I need to show that (i) $f_{n}(x)=\frac{1}{x^2+n^2}$ converges to the zero function in sup norm, and (ii) $g_{n}(x)=\frac{2nx}{x^2+n^2}$ does not. Not sure if this is right but would appreciate ...
1
vote
2answers
86 views

Textbook has wrong answer? - Metric spaces “topological properties” (probably trivial for the confident)

In the book there's a table and above it it reads "we have crossed out the wrong answer" meaning the remaining one is right. I dispute this, there are 4, I thought I got the first one right, but I ...
1
vote
0answers
82 views

Prove that every compact metric space $K$ has a countable base, and that $K$ is therefore separable.

Prove that every compact metric space $K$ has a countable base, and that $K$ is therefore separable. How does the following look? Proof: For each $n \in \mathbb{N}$, make an open cover of $K$ by ...
0
votes
1answer
21 views

Showing that a subset of $\Bbb R^3$ defined by an inequality is open

I have a set, which I'm pretty sure is open and here is why I think so. Let $S = ${$(x,y,z) \in \mathbb{R}^3 \colon \frac{e^{x+y^2-z}-1}{x^2+y^2-z^3} > 7$} Now I know that $(0,0,0)$ can't be in ...
0
votes
1answer
36 views

Unable to show a set is open/closed

Im struggling with how to show a set is open or closed.. worse is i have a test on this in a few days.. Here's an example, let $S = [(x,y) \in \mathbb{R}^2 \colon \frac{x}{y} \leq 7 ] $ I have to ...
3
votes
0answers
156 views

Prove that every separable metric space has a countable base.

A collection $\{V_\alpha \}$ of open subsets of $X$ is said to be a base for $X$ if the following is true: For every $x \in X$ and every open set $G \subset X$ such that $x \in G$, we have $x \in ...
4
votes
0answers
85 views

Prove that $\mathbb{R}^k$ is separable

I'd like to show that $\mathbb{R}^k$ is separable. (A metric space is called separable if it contains a countable dense subset.) Here's what I have and I'd like to confirm with everyone to see if ...
4
votes
1answer
45 views

$f^{n_i}(x)\to y$ implies $f^{-n_i}(y)\to x$?

Let $(X, d)$ be a compact metric space and $f:X\to X$ be a homeomorphism. If there exists a sequence $n_i$ such that $n_i\to\infty$ as $i\to\infty$ and $x, y\in X$ are such that $f^{n_i}(x)\to y$ as ...
3
votes
1answer
46 views

Analysis question.

Is this set compact? $\{(x,y) \in \mathbb R^2 : |x|+|y|\leq 1\}$. I know that is closed and bounded so compact but I don't know how to show it is closed and bounded mathematically. This is the graph ...
0
votes
0answers
115 views

Metric-space, counterexample in Arzela-Ascoli Theorem

My book has very few examples, so I would like an example covering this. The theorem is stated as follows. "Let $(X,d_{X})$ be a compact metric space. A subset K of $C(X,\Re^{m})$ is compact if and ...
2
votes
1answer
59 views

Is there a metric space and meanwhile a linear space such that vector addition discontinuous but scalar multiplication operation continuous?

Some special problems about topological groups or topological linear space theory. Recently I have done some study in some respects about topological group, topological linear spaces. And I found it's ...
5
votes
1answer
104 views

Is every $G_\delta$ set the set of continuity points of some function $f$?

I can prove that given a function $f:X \rightarrow Y$, where $X,Y$ are metric spaces, the set $A \subseteq X$ of points on which $f$ is continuous, is $G_{\delta}$. (Take $U_n = \bigcup_{y \in ...
1
vote
1answer
64 views

Is there any non-translation invariant but homogeneous metric linear space?

A metric linear space is a metric space and vector space, and linear operation is continuous regarding to the metric. I know that a homogeneous, translation invariant metric $d$ can be used to define ...