0
votes
2answers
37 views

Product metric spaces is again a metric space

Let $(X,d_X)$ and $(Y,d_Y)$ be metric spaces, and let: $$ d_2 ((x_1,y_1),(x_2,y_2)) = \left[d_X(x_1,x_2)^2 + d_Y (y_1,y_2)^2 \right]^{\frac{1}{2}} $$ for the points $(x_1,y_1)$ and $(x_2,y_2)$ in $X ...
1
vote
1answer
23 views

Mapping on induced topology and distance metric

Let $(X, d)$ be a metric space. Let $τ$ be the metric topology on $X$ induced by $d$. For $A ⊆ X$ , let $d(x, A) := \inf_{a∈A} d(x, a) $ for $x ∈ X$ (a) If $f (x) := d(x, A)$ (for a fixed subset ...
0
votes
0answers
29 views

About definition of separable metric spaces

My professor says in class that a metric space $(X,d)$ is separable if exist a countable dense subset $D$, $X$ is a subset of the set of limit points of $D$. I think it seems not right. I think $X$ ...
1
vote
1answer
72 views

Relation between the covers by sets of small diameter and the size of uniformly separated sets

Sorry I didn't find a better title. Here is the problem and my solution so far, I'd appreciate if someone could told me if is correct and for the last point, which at first sight seems to be ...
1
vote
1answer
14 views

Completeness of $C_{X,\mathbb{C}}$

If I haven't committed any error in my proof, the space of continous applications mapping a compact space $X$ into $\mathbb{C}$ or $\mathbb{K}$ is complete with the metric defined by ...
2
votes
2answers
54 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
33 views

triangle inequality for a metric space

If $d_{\infty}(a,b) =$ max$\{|a_{i} - b_{i}|\}$ for $1 \leq i \leq k$, I want to prove that this is a metric on $\mathbb{R}^k$. Its pretty clear that $d_{\infty}(a,a) = 0$ and it is also pretty clear ...
0
votes
0answers
28 views

Inner product spaces, normed spaces, metric spaces and topological spaces

I am collecting theorems or properties that hold in IPS, NS, MS or topological spaces, but not all of them. The reason is that I want to create some sort of overview over the respective spaces and ...
0
votes
2answers
47 views

Prove that the space of sequences under this metric is complete and compact.

I'm currently studying for the prelim exams, and I would love a hint on how to complete this problem. If $X$ is the space of sequences of $0$'s and $1$'s (i.e., $x \in X$ if $x = (x_{1}, x_{2}, ...
0
votes
2answers
39 views

Continuous function vs Uniformly continuous function

Can you give me an example of the function in metric space which is continuous but not uniformly continuous. Definitions are almost the same for both terms. This is what I found on wiki: ''The ...
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
22 views

Prove that $p \mapsto \inf\{d(p,s) : s \in S\}$ is uniformly continuous

From Pugh's Real Mathematical Analysis, #14, p. 115: The distance from a point $p$ in a metric space $M$ to a non-empty subset $S \subset M$ is defined to be $\text{dist}(p, S) = \inf\{ d(p,s) : s ...
2
votes
2answers
31 views

Problem showing that $\partial D = \emptyset$

I am considering $(X,d)$, which is a set with a discrete metric and $D \subset X$. With these conditions given, I am supposed to show that $\partial D = \emptyset $. Further I am supposed to describe ...
0
votes
0answers
26 views

Some neigbourhood properties

I am given an exercise where I am supposed to show some properties of neighbourhoods. I am given a metric space ($X$,$d$) and two points, $p$ and $q$, that both are members of $X$ ($p$,$q$ $\in$ ...
0
votes
0answers
44 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 ...
1
vote
1answer
39 views

Tails sets are Borel

I am trying to proof a particular case of Kolmogorov's law in the set E of infinite binary sequences. Eventually, I'm supposed to prove that a certain type of subsets of this set is in the Borel sigma ...
0
votes
2answers
58 views

Bounded metric on $\mathbb{R}$

I am supposed to give an example of a metric that is bounded on $\mathbb{R}$. In other words, I have to find a function $d:\mathbb{R} \times \mathbb{R} \rightarrow \mathbb{R}$ which satisfies that $d ...
0
votes
1answer
30 views

Two definitions of compact set

I am reading parallely two books on analysis, and they have two different definitions of compact set: 1) Subset A of metric space X is called compact, if every open cover of A contains a finite ...
0
votes
3answers
37 views

Proving that a function on a compact metric space is bounded above

The question is as follows. Let $(X,d)$ be a compact metric space, and let $f:X \rightarrow \mathbb R $. Assuming that for each $r \in \mathbb R$, the set $G_r=\{x \in X : f(x) \lt r\}$ is open, prove ...
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 ...
1
vote
1answer
17 views

Is this theorem about “completion of metric space” correct?

It's well-known that there is a completion of a metric space unique upto isometry. I have tried to modify this theorem slightly and I proved this statement: Let $(X,d_X)$ be a metric space. ...
1
vote
0answers
29 views

$p$-adic metric proof

I need to prove this, Let $p$ be an odd number. It is defined the function $v_p:\mathbb{Q}\to \mathbb{Z}$ as $$v_p\left(p^n\frac{a}{b}\right)=n, \hbox{ if } \mathrm{mcd}(a,p)=\mathrm{mcd}(b,p)=1.$$ ...
1
vote
2answers
46 views

Existence of a metric space M with no continuous map from M to any other metric space

Is it possible to have a metric space M such that there is no continuous map from M to any other metric space?
1
vote
1answer
23 views

Unit ball of continuous functions is a closed set - Proof with neighborhood argument

This question is trivial if one uses sequence definition, but I want to use the usual topological definition of closed set. That is , a set is closed if its complement is open. Let $U=\{f\in ...
1
vote
1answer
43 views

Choice of Metric Gives Nice Topological Properties

I am looking for examples where choosing one possibility out of many for a metric gives nice topological properties compared to the other choices. Nice is defined as compact, Hausdorff, or whatever ...
3
votes
1answer
95 views

Cauchy sequence in compact metric space

Suppose $f:X\rightarrow X$ continuous function, $X$ is compact metric space with $\rho(f(x),f(y))<\rho(x,y)$ for any $x\neq y$. Let $x_n=f(x_{n-1})$, with $x_0\in X$ arbitrary. I want to show that ...
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 ...
1
vote
2answers
34 views

Compact sets and Kuratowski limit

I have been struggling with the following claim: Let $A_n$ be a sequence of compact sets and $A$ a compact set. $A=\lim\sup_n A_n=\lim\inf_n A_n$ iff $d_H(A_n,A)\to 0$ where $d_H(.,.)$ is the ...
2
votes
1answer
39 views

Discontinuous function whose restriction on closed sets is continuous

Let $X$ a metric space, $\{U_i\}$ a collection of non-empty closed sets whose union is all of $X$. Give an example of a function $f:X\rightarrow \mathbb{R}$ such that the restriction $f|_{U_i}$ is ...
2
votes
3answers
104 views

Compactness under different metric?

Consider the metric $\rho(x,y)=\frac{|x-y|}{1+|x-y|}$ on $\mathbb{R}$. Is $(\mathbb{R},\rho)$ compact? In order to show that is not, I wanted to find a sequence such that any subsequence is ...
0
votes
1answer
29 views

Why is this set closed?[metric-spaces]

I am reading a note, where part of it is this: Why is S' closed? I have tried to argument like this, but I am not able to finish the argument: Let $\{x_n\}$ be a convergent sequence in S', then ...
1
vote
2answers
214 views

How do I turn my verbal argument into something formal in [Real Analysis]? (proving every compact set is bounded)

So one of the exercises I am doing is to prove (or disprove) that 'Every compact set on a metric space is bounded'. Verbally, I can 'prove' this by simply stating: "If the every compact set on a ...
0
votes
0answers
36 views

How do I turn my verbal argument into something formal in Real Analysis? [duplicate]

So one of the exercises I am doing is to prove (or disprove) that 'Every compact set on a metric space is bounded'. Verbally, I can 'prove' this by simply stating: "If the every compact set on a ...
2
votes
4answers
147 views

How to finish this proof about compact implies bounded

A set is called compact if every sequence has a convergent subsequence. I am trying to show: If $K$ is compact then it is bounded. (that it is closed was very easy to prove) What I want to do: Let ...
1
vote
1answer
30 views

Lipschitz continuous set-valued map

Let $X$ and $Y$ be metric spaces and $F:X\to2^Y$ be a set-valued map. Suppose that $2^Y$ is endowed with the Hausdorff metric. I wonder about sufficient conditions on $F$ that ensure this map is ...
1
vote
2answers
60 views

We know that a Compact set is closed. However a finite discreet set is compact but not closed (contradicting the theorem?)

We know that a Compact set is closed. We also know that a finite discreet set is compact (as every cover has a finite sub cover). However a finite discreet set is not closed (contradicting the ...
0
votes
1answer
20 views

metric-spaces closure

The line in my writing symbolizes the closure, where we add all the boundary points. Is it true that: $\overline{A \cap B}=\overline{A}\cap {B}$? If B is closed, but we do not know if or if not A ...
2
votes
1answer
43 views

Understanding a statement about equivalent norms ($||\cdot ||_2 \sim||\cdot||_1 $)

I am trying to understand the following statement from an analysis book: Two norms are equivalent ($||\cdot ||_2 \sim||\cdot||_1 $) if they induce equivalent metrics. At first I thought this ...
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 $(X_2, d_2)$ ...
0
votes
1answer
25 views

Condition for Lipschitz functions and set inclusion

Let $(X,d)$ be a pseudometric space, and for each $A\subseteq X$ and $\varepsilon \geq 0$ define $$ A^\varepsilon := \{x\in X:\exists a\in A \text{ s.t. }d(x,a)\leq \varepsilon\}. $$ Is this set ...
3
votes
1answer
62 views

Qualifier problem: Completeness of Metric Spaces

I am working on old qualifier problems as a review, and I came across this one: Suppose there exists a continuous surjection $f:X_1 \mapsto X_2$, where $(X_1,d_1),(X_2,d_2)$ are metric spaces, such ...
0
votes
2answers
50 views

if $A, B$ are open in $\mathbb R$ then so is $A+B.$

I am trying to find out a counterexample to the problem: if $A, B$ are open in $\mathbb R$ then so is $A+B.$ But I could not find any such counterexample. Please help me.
1
vote
1answer
44 views

How to check whether this function is continuous or not..?

Let A and B be two disjoint closed sets of any Metric space X.I have to construct a continuous function such that $f(x):= 0$ if x belongs to $A$ $f(x) = 1$ if x belongs to $B$ My idea is to use the ...
1
vote
0answers
46 views

Characterization of continuity by subsequences.

I have a difficulty trying to prove the following proposition. Any help would be greatly appreciated. $\textbf{Prop.}$ Let $(X,d_1)$ and $(Y,d_2)$ be two metric spaces. A function $f:X\rightarrow Y$ ...
0
votes
0answers
37 views

Verifying the triangle inequality for a metric for hyperbolic space

I read that the formula $d(x,y)=\mathrm{arccosh}(1+\frac{2||x-y||^{2}}{(1-||x||^{2})(1-||y||^{2})})$, where $x,y$ are in the open unit ball of $\mathbb{R}^{n}$ and $||\cdot||$ denotes Euclidean norm, ...
0
votes
3answers
47 views

Which of the following sets are dense in $C[0,1]$

Which of the following sets are dense in $C[0,1]$ with respect to sup-norm topology? $1$. {$f$$\in$ $C[0,1]$ : $f$ is a polynomial } $2$. {$f$$\in$ $C[0,1]$ :$f(0)$=$0$} $3$. {$f$$\in$ $C[0,1]$ ...
0
votes
2answers
26 views

replace convergence with continuity?(metric spaces)

This question is convcerning metric-spaces. In theory we can replace continuity with convergence. That is, since continuity in a point a is equal to the statement that if $\{x_n\}$ is any sequence ...
1
vote
1answer
43 views

Let $f$ be a continuous functions from $[0,1]$ to $\mathbb{R}$. Then, $f$ is not necessarrily lipschitz.

Let $f$ be a continuous functions from $[0,1]$ to $\mathbb{R}$. Then, $f$ is not necessarily lipschitz. Is the above statement true? I thought since $f$ is continuous on a compact metric space, $f$ ...
1
vote
1answer
39 views

Can I show these questions (is a set open or closed WRT metric) a faster way?

I have the metric: $$d((x,y),(a,b))=|y-b|\text{ if }a=x\text{ else }|y|+|b|+|x-a|$$ I have been asked the following questions: Is the set $\{0\}\times(0,1)$ open with respect to this metric? Is it ...
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 $d_1(x, y) \le C\, d_2(x, y)\ \ \ \forall x, y \in X$. ...