0
votes
1answer
22 views

Prove that the distance function $d_p(x,y)=\sum_1^n |x_i-y_i|^p$ $0<p<1$ is a metric on R^n

Hi I am trying to prove that for $0<p<1$ the function $d_p(x,y)=\sum_1^n |x_i-y_i|^p$ is a metric on $\mathbb{R}^n$. I am struggling with the triangle inequality part; We have to prove ...
0
votes
1answer
66 views

Is it true that $0<0$ is a part of the proof for a unique fixed point of a contraction map?

I think it isn't, but my friend told me that this statement is true, he said that $0<0$ is even part of the proof for a unique fixed point of a contraction map. I google it but I couldn't find ...
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 ...
1
vote
3answers
63 views

proving topological equivalence

How would I show that $$d_E = \sqrt{\sum_{i=1}^n(x_i-y_i)^2}$$ and $$d_\infty= \sum_{i=1}^n|x_i-y_i|$$ and $$\sup\{|x_1-y_1|,|x_2-y_2|,...,|x_n-y_n|\}$$ are topologically equivalent on $\mathbb{R}^n$? ...
1
vote
2answers
57 views

proving a metric

I'm trying to show that given a metric $d(x,y)$ show that $d_0(x,y) =\frac{d(x,y)}{1+d(x,y)}$ is also a metric.. It's trivial to show the first two properties, that is, $d_0\geq 0 $ & for $x=y ...
1
vote
1answer
48 views

Open Subsets of open sets

How does one go about proving/disproving that given $(X,d)$ a metric space that a subset $S$ is open. Given the following definitions: A set $X$ is open $\iff \forall x \in X, x\in int(X)$ i.e. x ...
-1
votes
2answers
58 views

Equivalence Classes

The scenario is a follows: I am given a set $X$ along a map $d$ defined as $d:X \times X \to \Bbb R^+$ for all $x,y,z \in X$ with the following properties: $d(x,y) =0 \Leftarrow x =y \\ d(x,y) ...
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 ...
0
votes
2answers
102 views

Two equivalent definitions of convergent sequences?

I know that: Definiton 1. The sequence $(x_n)$ in the metric space $(X,d)$ is said to converge to the point $x_0\in X$ if $$\forall\epsilon>0, \exists n_0\in\mathbb{N} \text{ such that } \forall ...
0
votes
1answer
78 views

Connected subsets of a metric space

I have to prove the following result: Suppose $X$ is a metric space, $Z$ is a metric subspace of $X$ and $S \subset Z$ Then $S$ is a connected subset of $X$ if, and only if,$S$ is a connected subset ...
0
votes
1answer
59 views

If $X, d$ is a connected metric with at least two points, prove it is uncountable. [duplicate]

I was able to prove the theorem by contradiction, showing that if $X$ is a countable set, we can take $r_0 = min\{d(p, q) < r : \forall p, q \in X\}$ and so there exists an $N_r(p)$ such that it ...
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 ...
0
votes
2answers
39 views

$A \subseteq (X,d)$ is compact. Which metric $p$ makes $(A \times A,p)$ also compact and $d: (A \times A,p) \rightarrow [0,\infty)$ continuous?

$(X,d)$ is a metric space. And $A \subseteq X$ is a non-empty compact set in the metric space $(X,d)$. Then, does there exists a metrics $p$ and if so which metrics $p$ make $(A \times A,p)$ compact ...
2
votes
1answer
70 views

Help with proof

Let $X$ be a metric space, $x \in X$ and $S \subset X$. Then, I have to prove that $x \in Cl(S)$ if and only if, every open ball of $X$ centered at $x$ has non-empty intersection with $S$. I managed ...
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 ...
2
votes
0answers
62 views

Is this case possible (hedgehog metric, colinearity)

My topology class was asked to prove that the hedgehog metric was indeed a metric (the details are irrelevant for my question). This does not concern the proof itself, but rather the structure of the ...
3
votes
3answers
245 views

Metric spaces and openness

I am asked to prove two things. I would like to know if the proof was elaborate and concise. I would also like to know if proving reductio ad absurdum is looked down upon. I have heard from my ...
3
votes
2answers
98 views

Using axioms to define metric spaces

Let $M$ be a set with three elements: $a$, $b$, and $c$. Define $D\colon M\times M\to[0,\infty)$ so that $D(x, x) = 0$ for all $x$, $D(x, y) = D(y, x)$ for $x \ne y$. Say $D(a, b) = r$, $D(a, c) = s$, ...
2
votes
5answers
248 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
978 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 ...
2
votes
1answer
87 views

Prove equivalence of definitions of “dense”

Prove that these two statements are materially equivalent (that is, one statement can be derived or proven from the other). Read below the statements for background information if you need it. Given ...
0
votes
1answer
91 views

Let $X$ be a Moore space and $e(X)=\omega$. Is it metrizable?

Let $X$ be a Moore space and $e(X)=\omega$. Is it metrizable? What I've tried: I list these facts: 1 A space $X$ is a Moore space iff $X$ is a $\sigma$-space and a $p$-space. 2 If $X$ is a ...
3
votes
2answers
57 views

Any practical difference between the metrics $d_1=\sup\{\left|{x_j-y_j}\right|:j=1,2,…,k\}$ and $d_2=\max\{\left|x_j-y_j\right|:j=1,2,…k\}$?

I've been asked to do a proof showing that $d_1\left({x,y}\right)$ is a metric on $\mathbb{R}^k$, but is there any difference between this and $d_2\left({x,y}\right)$, for which I've already done a ...
1
vote
1answer
76 views

Prove that $d_2=\sum_{j=1}^{k}\left|{x_j-y_j}\right|$ is a complete metric on $\mathbb{R}^k$.

I am trying to prove that $d_2\left({x,y}\right)=\sum_{j=1}^{k}\left|{x_j-y_j}\right|$ is a complete metric on $\mathbb{R}^k$. Here is my reasoning for why it is, but I am unsure about its ...
3
votes
3answers
57 views

Prove that $d_1\left({x,y}\right) = \max\{\left|{x_j-y_j}\right|:j=1,2,…,k\}$ is a metric on $\mathbb{R}^k$

I am trying to prove that $d_1\left({x,y}\right) = \max\{\left|{x_j-y_j}\right|:j=1,2,...,k\}$ is a metric on $\mathbb{R}^k$. So far I know that $\text{dist}\left({x,y}\right) \leq ...
1
vote
0answers
116 views

Determining Complete Metric Spaces

I need to determine if $((0, 1), d)$ where $d(x, y) = |x^2 - y^2| \forall_{x,y}\in (0,1)$ My argument is as follows, take the sequence defined by $\dfrac{1}{n}$, then we know by $d(x, y)$ that ...
0
votes
2answers
67 views

Does this proof make sense and correct — is it written well enough?

I'm working on a tutorial question. The question asks whether the following claim is true or false, if it is true: one is supposed to provide a proof or counter-example otherwise if it's false. Let ...
1
vote
1answer
466 views

Show $\mathbb R^n$ is complete.

Show $\mathbb R^n$ is complete. At this point, I am trying to work through the problem in my textbook, there is one step that I do not understand and would like explained. Here's my proof so far: ...
0
votes
0answers
82 views

What's wrong with my proof that a continuous function is uniformly continuous?

I was trying to prove that any continuous function from a compact metric space to any other metric space is uniformly continuous. I proved it as follows: Let $f\colon X\to Y$ be continuous and ...
6
votes
3answers
1k views

'Every open set in $\mathbb{R}$ is the union of disjoint open intervals.' How do you prove this without indexing intervals with $\mathbb{Q}$?

In my book's exercises section I am asked to prove that every bounded, open set in $\mathbb{R}$ is the union of disjoin open intervals. Looking around the internet I have found many strategies that ...
0
votes
2answers
168 views

Metric spaces question

\begin{equation} P = \{f \in\ C ( \mathbb{R}, \mathbb{R}) \mid f(x+2 \pi ) = f(x)\} \end{equation} be the set of $2\pi$-periodic function. 1) Show that $P$ is a subspace of $C( \mathbb{R}, ...
4
votes
4answers
126 views

Show that the set given is closed

A question that I encountered which looks different than a normal open/closed sets proofs: Let $(E, d)$ be a metric space, let $f : E\to R$ be continuous and $a$ element of $R$. Show that the set ...
3
votes
4answers
502 views

Show that the set is closed

Let $(E, d)$ be a metric space, $x$ element of $E$. Show that the set \begin{equation} A = \{y \in\ E : d(x, y) \geq 5 \} \end{equation} is closed. Generally, how would you go about this? I have an ...
2
votes
1answer
404 views

Topology: Proving a space is connected

I'm attempting to prove that a space is connected and compact. I have a continuous function $f:X \rightarrow S^{1}$. $X$ is metrizable and locally connected. $f$ is non-constant, surjective and ...
3
votes
1answer
286 views

On two different proofs

This question is mainly to understand the meaning of my professor's correction to a proof of a theorem I gave during an oral examination. The question was to show that $\text{diam}A = ...