0
votes
0answers
16 views

Proof verification related to the discrete metric

Can someone please verify my proof? Let $X_1$ be a set and let $d_1$ be the discrete metric on $X_1$. (a) Prove that every subset of $(X_1, d_1)$ is open. (b) Prove that if $(X_2, d_2)$ ...
4
votes
2answers
75 views

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 ...
0
votes
1answer
31 views

Can someone criticise my incorrect proof about a set being open?

In the question I have to decide whether the set $S=\{(x,y)\in\mathbb{R}^2\;|\;x/y\leq 7\}$ is open, closed or neither. I attempted to prove it was closed but it turns out it is neither can someone ...
2
votes
1answer
32 views

Cauchy inequality proof

I am studying cauchy inequality proof from notes I have from my class$$(\forall\vec{x},\vec{y}\in\mathbb{R}^n):|\sum_{i=1}^{n}x_iy_i|\le||\vec{x}||\cdot||\vec{y}||$$ We choose $\vec{x},\vec{y}$. And ...
1
vote
1answer
27 views

About interior of the frontier (proof-checking)

Let $M$ be a metric space, and $A \subset M$ an open set. Show that $\stackrel{o}{\widehat{\partial A}} = \emptyset$. ($\stackrel{o}{\widehat{\partial A}}$ is the interior of the frontier) I ...
0
votes
2answers
54 views

Show graph is closed if f is continuous

I need help figuring out how to complete this proof: Show that the graph of $f$ is closed if $f$ is continuous.
1
vote
2answers
66 views

Proving a metric induces the product topology

Let $(M,d)$ and $(N,d')$ be metric spaces. Prove that the product topology is induced by the metric $d_1((x,y),(x',y')=d(x,x')+d(y,y')$ and ...
1
vote
0answers
16 views

Creating a metric from a pseudometric

Given the following definition of a pseudo-metric on the set $X$ : A pseudo-metric on the set $X$ is a map $d:X \times X \to \Bbb R^+$ such that for all $x ,y \text{ and } z \in X :$ (PM1) $x=y ...
0
votes
0answers
18 views

The Discrete Metric

I feel rather embarrassed and ashamed asking this question , since it is so obvious. Anyway here I go... I am trying to proof rigorously that the Discrete Metric , which is defined as: $$ ...
1
vote
0answers
27 views

Proving $d_E$ is a Metric on $\Bbb R^n$

This question rather then wanting help w.r.t (M3) (The Triangle Inequality ) with can be done by using the Cauchy–Schwarz inequality. I would like advice regarding (M1): $d_E(\mathbf {q,p}) = 0 \iff ...
0
votes
1answer
28 views

The Open Set $X-\lbrace x \rbrace$

I am task with proving the following: if $x \in X$ then $X- \lbrace x \rbrace $ is an open set I kind of have an idea but I am unsure about it and how to express it. I was thinking about using the ...
3
votes
4answers
48 views

(Proof Checking) Prove that if $C,C'$ are compact subsets of a hausdorff space $X$ then $C\cap C'$ is compact in $X$.

Prove that if $C,C'$ are compact subsets of a hausdorff space $X$ then $C\cap C'$ is compact in $X$. I am tempted to use the following argument. Let $U = \{U_i|i\in I\}$ be some open cover of $C\cap ...
3
votes
1answer
71 views

The “intersection property” of the symmetric difference metric

$\newcommand{\measure}{\operatorname{measure}}$ The symmetric difference between sets can be used to define a pseudo-metric on the set of subsets of a given measure space: $$d(S,T)=\measure(S\oplus ...
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 $x-y=0=p^0\dfrac0n$. ...
2
votes
1answer
156 views

Hausdorff distance and intersection

The question is related to the Hausdorff distance between sets, $d_H(S,S')$, which is the greatest of all the distances from a point in one set to the closest point in the other set. Suppose there ...
1
vote
1answer
51 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 ...
2
votes
1answer
72 views

Metric Spaces: The dist function

Given that $A$ is defined as non-empty subset of $(X,d)$ The distance function is defined as such: $dist(x,A)=$ inf $_{y\in A} \lbrace d(x,y) \rbrace $ Given the above we are asked to prove the ...
3
votes
0answers
57 views

Proof about sequences of functions.

Is this proof correct? If $\{f_{n}\}$ is a sequence of functions in $C(X,Y)$, $X$ compact, $Y$ complete, and the sequence converges, to $f$, then $K=(\bigcup\{f_n\})\cup \{f\}$ is closed. Proof. ...
1
vote
0answers
35 views

Metric space, I would like to rewrite this proof, completeness and compactness

I am looking at metric spaces. For the metric space C(X,Y) the metric is: $\rho (f,g)=sup\{|f(x)-g(x)| :x \in X\}$ This is the proof I am reading about. I would like to do the proof without just ...
1
vote
0answers
86 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 ...
3
votes
0answers
161 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 ...
2
votes
4answers
58 views

$E$ Closed iff $\partial E \subseteq E$

I'm having trouble verifying my proof, would appreciate some input on this one. Let $(X,d)$ be a metric space with $E\subset X$. Suppose $E$ is closed in $X$, which means that $E=\overline{E}$. By ...
1
vote
1answer
37 views

Is this proof correct - simple proof about distance in normed linear space

Let $X$ be a normed linear space, and let $M$ be a closed subset of $X$. Show that any $x\in X\setminus M$ has non-zero distance from M. My proof: Assume there exists an $x\in X$ such that ...
0
votes
2answers
36 views

Let $X$ be a metric space and $K$(Compact), $C$(Closed) $⊂X$ such that $K∩C=∅.$Show that $d(K,C)>0.$

Please tell if my proof works for the following problem: Let $X$ be a metric space and $K$(Compact), $C$(Closed) $⊂X$ such that $K∩C=∅.$Show that $d(K,C)>0.$ SOLUTION: $d_C:X\to\mathbb ...
0
votes
1answer
140 views

The distance between two disjoint compact subsets $A,B$ of a metric space $X$ is positive

Please tell me whether my argument for the following result is true: The distance between two disjoint compact subsets $A,B$ of a metric space $X$ is positive: $d:X\times X\to \mathbb R$ is ...
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
votes
1answer
47 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 + ...
1
vote
0answers
27 views

Proof that compactness can be characterized by closed sets.

If anybody would be willing to check to see if this proof is correct I would really appreciate it. Prove that a metric space $(X,d)$ is compact if and only if for any family $(C_i)_{i \in I}$ of ...
1
vote
1answer
84 views

Property of a metric in the space of all the sequences of real numbers

A few weeks ago I had this problem, the adjoint-teacher solved it on class, and I thought I understood, but now I'm rechecking and there are a few things that aren't clear for me. So we defined this ...
1
vote
1answer
254 views

Proving that Union of a finite number of complete subsets of metric space $(X,d)$ is complete.

We have following 3 definitions. Definition: Suppose that $(X,d)$ is a metric space. A sequence $(\textbf{x}_n)_{n\in \mathbb{N}}$ of points in $X$ is said to be a Cauchy sequence, if, given any ...
0
votes
1answer
76 views

Proof on if a set is discrete

I would like to know how well I answered the following proof: was it concise? Was it elaborate/rigorous? Did I use incorrect notation? I would also like to know if the set is a $T_{1}$ space, such ...
2
votes
2answers
455 views

Is this proof that all metric spaces are Hausdorff spaces correct?

Let $x$ and $y$ be distinct points of a metric space $M$. Prove that there exist in $M$ disjoint open sets $U$ and $V$ with $x \in U$ and $y \in V$. Let $U$ and $V$ be open balls centered at $a$ and ...
0
votes
1answer
58 views

Proof of the Banach Fixed Point Theorem

I am presenting to you my book's verion of the proof of the Fixed Point Theorem: $\{T^i x_0\}$ has been shown to be a cauchy sequence. As we have a complete metric space, this cauchy sequence has a ...
-1
votes
2answers
34 views

Ultra-metricity in sets

Suppose there is a set $A$ equipped with a trivial distance function. Take $D(a, a) = 0, D(a, b) = 1, a \not= b \in A$. A set $A$ is ultra-metric if $D(a, c) \le \max[D(a, b), D(b, c)]$ for all $a, ...
4
votes
2answers
110 views

Is this proof of uniqueness of the limit correct?

I've tried to show the following: let $(M,d_M)$ and $(N,d_N)$ be metric spaces and $f : M \to N$. If $a \in M$ and $\lim_{p\to a}f(p)$ exists, then it is unique. I'm a little unsure if the proof is ...
2
votes
1answer
60 views

Prove that the function $d_A:X\to\mathbb R:x\mapsto\displaystyle\inf_{a\in A} d(x,a)$ is continuous.

For a metric space $(X,d)$ and a nonempty subset $A$ of $X$ prove that the function $$d_A:X\to\mathbb R:x\mapsto\displaystyle\inf_{a\in A} d(x,a)$$ is continuous. Choose $c\in X$ and ...
2
votes
5answers
249 views

Let $(X,d)$ be a compact metric space. Prove that there exists a number $K$ such that $d(x,y)\leq K$ for each $x,y\in X$.

I'm reading Intro to Topology by Mendelson. The problem statement is in the title. My attempt at the proof is: Since $X$ is a compact metric space, for each $n\in\mathbb{N}$, there exists ...
2
votes
1answer
982 views

Prove that a compact metric space is complete.

I'm reading Intro to Topology by Mendelson. I'm in the section titled "Compact Metric Spaces". The problem is in the title. My attempt at the proof is as follows: Let $\{a_n\}_{n=1}^\infty$ be a ...
3
votes
1answer
118 views

$O(n,\mathbb R)$ of all orthogonal matrices is a closed subset of $M(n,\mathbb R).$

Let $M(n,\mathbb R)$ be endowed with the norm $(a_{ij})_{n\times n}\mapsto\sqrt{\sum_{i,j}|a_{ij}|^2}.$ Then the set $O(n,\mathbb R)$ of all orthogonal matrices is a closed subset of $M(n,\mathbb ...
2
votes
0answers
291 views

Is this proof that every convergent sequence is bounded correct?

I've tried the following proof that a convergent sequence is bounded but I'm not sure if it is correct or not. Let $(M,d)$ be a metric space and suppose $(x_k)$ is a sequence of points of $M$ that ...
1
vote
1answer
65 views

Show that for $n\in\mathbb N$ the mapping $f:M(n,\mathbb R)\to M(n,\mathbb R):A\mapsto A^n$ is continuous.

Show that for $n\in\mathbb N$ the mapping $f:M(n,\mathbb R)\to M(n,\mathbb R):A\mapsto A^n$ is continuous. ($M(n,\mathbb R)$ is identified with $\mathbb R^{n^2}$ as a normed liner space.) ...
2
votes
0answers
71 views

Let $x=(x_n)$ be a Cauchy sequence in a metric space $X$ having a convergent subsequene

Let $x=(x_n)$ be a Cauchy sequence in a metric space $X$ having a convergent subsequene $x\sigma=(x_{\sigma_n})$ where $\sigma:\mathbb N\to\mathbb N$ is strictly increasing. Then $(x_n)$ is ...
2
votes
1answer
355 views

Justify: In a metric space every bounded sequence has a convergent subsequence.

Justify: In a metric space every bounded sequence has a convergent subsequence. My Attempt: False: Consider the metric space $(X,d)$ where $X=\mathbb R$ and $d$ is the discrete metric on $X.$ ...
0
votes
1answer
175 views

Properties for interior and closure in metric space.

I found the some following properties for general topology and prove these. But, I want to verify that the proofs are really true. Let $(X,d)$ be metric space. Let $A$ be any subset of $X$. Define ...
1
vote
1answer
100 views

Show that the diagonal $\{(x,x): x\in X\}$ is closed in the metric space $(X\times X,d=\max\{d_X,d_X\})?$

Show that the diagonal $\{(x,x): x\in X\}$ is closed in the metric space $(X\times X,d=\max\{d_X,d_X\})?$ My attempt: Choose $(x,y)\in X\times X-\{(a,a): a\in X\}$ Then $c=d(x,y)/2>0.$ To ...
3
votes
1answer
49 views

Why do the author added the extra condition that $X$ needs to be $T_1?$

In my text it's written that, But I get to prove the result underlined red simply for a first countable space as: (N.B. by limit point the author wanted to mean the adherent point) ...