0
votes
0answers
44 views

How to show that addition is continuous?

Let $f: R \times R \rightarrow R$ and let the metric over $R$ be $d(x,y)=|x-y|$ and let the metric in $R \times R$ be $d_2((x,y),(a,b))= ((x-y)^2+(a-b)^2)^{1/2}$. I believe I understand how to ...
1
vote
2answers
17 views

Convergence of functions in a metric space

Let $C([0,1])$ be the space of all continuous functions from $[0,1]$ to $\mathbb{R}$ under the metric $$ \lVert f \rVert_1 = \int_0^1 \lvert f(x) \rvert \, dx. $$ Now consider $f_n(x) = e^{-nx}$. I ...
3
votes
2answers
85 views

Proving a set is compact - Homework

Let $(X,d)$ be a metric space and let {$p_n$} be a sequence of points in $X$ with $\lim_{n\to ∞}p_n = p_0$. Prove that the set $K =$ {$p_0, p_1, p_2,...$} is a compact subset of $X$. I have ...
1
vote
3answers
51 views

Distance function is in fact a metric

I know I should be able to show this, but for some reason I am having trouble. I need to show that $$d(x,y) = \frac{|x-y|}{1+|x-y|}$$ is a metric on $\Bbb R$ where $|*|$ is the absolute value metric. ...
4
votes
1answer
43 views

Continuous function with infimum

Let $A$ be a closed subset of a metric space $X$ and $f:A \rightarrow[1,2]$ a continuous function on it. Now I want to find out why the function $$F(t):=\frac{\inf\{f(s)d(s,t);s \in A\}}{\inf ...
2
votes
1answer
47 views

Closed sets in a subspace are formed by intersecting the subspace with closed sets

Let $X$ be a metric space and let $Y$ be a subset of $X$ be a subspace with the induced metric. (induced means the metric restricted to elements of $Y$) Let $A$ be a subset of $Y$. Prove the following ...
1
vote
1answer
16 views

Distance to a set

I have a question concerning to the following problem. Let $(X,d)$ be a metric set. For every subset $T \subset X$ we define a mapping \begin{equation} d_T : X \rightarrow R , d_T(x) := inf\{d(x,y) | ...
2
votes
1answer
42 views

MAth proof questions Open closed sets

Let $X$ be a metric space and let $Y$ be a subset of $X$ be a subspace with the induced metric. (induced means the metric restricted to elements of $Y$) Let $A$ be a subset of $Y$. Prove the following ...
2
votes
1answer
22 views

Why open unit ball in any infinite dimensional Banach space is finitely chainable?

In paper "Pointwise products of uniformly continuous functions" by Sam B. Nadler, Jr., He defined the finitely chainable as followings : Let $(X,d)$ be a metric space. An $\varepsilon$-chain in ...
1
vote
1answer
18 views

Approximation of acontinuous function

How to approximate a continuous function on $[-\pi,+\pi]$ which is $2\pi$ periodic by a set of trigonometric polynomials in the sup-norm topology?
2
votes
3answers
35 views

New metric spaces from given one

Let (X,d) be a metric space, $f : [0,\infty) \rightarrow [0,\infty)$ continuous differentiable, strict monotone increasingly with $f(0)=0$ and a monotone decreasingly derivative. Prove that $f \circ ...
0
votes
1answer
51 views

Proof of that there is no metric on $\mathbb{R}$ which is equivalent to the natural metric and which induces a metric on $(0,1)$

I want to prove the following statements: For $X:=(0,1),$ prove the following: (a) $d(x,y):=\big|(1/x)-(1/y)\big|$ is a metric on $X.$ (b) The natural metric and $d$ are equivalent. (c) There is ...
1
vote
1answer
117 views

Why isn't $C[0,1]$ a Banach space in this unusual norm?

I need to answer the following question: Let $X$ be the normed space $X=C([0,1])$, with norm defined as $$ \|\,f\|= \max_{x\in[0,1]} x^2 \lvert\,f(x)\rvert. $$ Why isn't this a Banach space?
-1
votes
3answers
33 views

Example of metric space that has same interior and closure as its complement? [closed]

Please provide an example for this : Consider a metric space X. Let S be a subset of X. Then S and S complement both have the same interior and the same closure.
0
votes
2answers
33 views

In a semi-metric space, need the limit of a sequence be unique as it is in a metric space? Yes

In a metric space (M,d), define what you mean by a bounded set B and by L being the limit of a sequence $\{x_n\} \in M$. What is the sequential definition of a function between two metric spaces being ...
2
votes
3answers
69 views

Prove that the distance between 2 Cauchy sequences is convergent.

Here is the exact question: Let $(S,d)$ be a metric space. Let $(p_n)$ and $(q_n)$ be two Cauchy sequences in $(S,d)$(note that these two sequences are not necessarily convergent since $(S,d)$ is ...
0
votes
2answers
32 views

In a metric space with a countable base, how does every open cover has a countable subcover?

Let $X$ be a mertic space, and let $\{V_{\alpha}\}$ be a collection open subsets of $X$ such that, for every $x \in X$ and for every open set $G \subset X$ with $x\in G$, there is some $V_\alpha$ such ...
2
votes
1answer
46 views

Closed Ball Complete iff $(M,d)$ is complete

I encountered the following in Carothers' Real Analysis: Prove that $(M,d)$ is complete iff for each $r>0$, the closed ball $B_r=\{y\in M: d(x,y)\leq r\}$ is complete. Attempt/Thoughts: ...
1
vote
3answers
38 views

Show $\ell_\infty (M)$ is a Banach Space

I'm working on problems from Carothers' Real Analysis. The following problem is in the section on completions. Given any metric space $(M,d)$, check that $\ell_\infty(M)$ is a Banach space. ...
4
votes
1answer
160 views

Completeness of the space of sets with distance defined by the measure of symmetric difference

Let $m$ be the measure defined on the set semiring $\mathfrak{S}_m$ and $m'$ its extension to the minimal ring $\mathfrak{R}(\mathfrak{S}_m)$. I read that $m'(A\triangle B)$ can be used as a distance ...
4
votes
3answers
32 views

Cauchy Sequence some challenge

i read this sentence in one of math books: ‌Every convergent sequence in metric space is a cauchy sequence. would you please some one add more detail, why this is true? thanks.
2
votes
3answers
59 views

If $X = \{x_n:n \in \mathbb N\}$ is a cauchy sequence in a metric space $S$ and $f : S \rightarrow T$ is continuous , is $f(x_n)$ a cauchy sequence?

If $X = \{x_n:n \in \mathbb N\}$ is a cauchy sequence in a metric space $S$ and $f : S \rightarrow T$ is a continuous function where $T$ is an another metric space , is $f(x_n)$ a cauchy sequence? ...
0
votes
1answer
44 views

Why are $\{0\}$ and $\{1\}$ open subsets of the discrete metric space $\{0,1\}$?

$\{0\}$ and $\{1\}$ open subsets of the discrete metric space $\{0,1\}$. Let $S$ be a metric space. Then a subset $A \subseteq S$ is considered open if $\forall x \in A, \exists r>0$ such that ...
0
votes
1answer
27 views

Discrete Sets in Certain Metric Spaces are Countable

I've recently started my first real analysis course, and we're studying metric spaces from Rudin's Principles of Mathematical Analysis (Baby Rudin). We have the following definition: Let $(X,d)$ be ...
0
votes
1answer
37 views

Let D be the usual Euclidean metric on M. Are the two identity functions $I_1$ and $I_2$ continuous functions where d is the Manhattan metric on M?

Let D be the usual Euclidean metric on M. Are the two identity functions $I_1:(M,D) \to (M,d)$ and $I_2:(M,d) \to (M,D)$ continuous functions where d is the Manhattan metric on M? The definition of ...
1
vote
1answer
25 views

Proving p is a Metric on X

I have an exercise which I cannot get my head around. Essentially $X$ is a non-empty set and $p: X^2 \to \mathbb{R}$ satisfies (1) $0 \leq p(x,y) < +\infty$ (2) $p(x,y) = 0 \iff x=y$ and (3) ...
2
votes
1answer
49 views

Is it possible to show that the addition of two Cauchy sequences in $\mathbb R^n$ is also Cauchy for any metric?

A problem in my homework asks to show that the addition of two Cauchy sequences in $\mathbb R^n$ is also Cauchy. However, the metric is not specified. If we assume that we are dealing with the ...
1
vote
2answers
59 views

Problem about metric spaces?

Let $X$ be an infinite set and let $d$ be the discrete metric on $X$. What sets in $X$ are open? Closed? Compact? Now, I know that $d$ will be either $0$ or $1$ since we are talking about the ...
3
votes
1answer
49 views

Is the space $B([a,b])$ separable?

Let $a$, $b$ be two real numbers such that $a < b$, and let $B([a,b])$ denote the metric space consisting of all (real or complex-valued) functions $x=x(t)$, $y=y(t)$ that are bounded on the closed ...
4
votes
1answer
26 views

continues function statement in real analysis [closed]

I ran into a challenge, i read following sentence in one note. anyone could describe or prove it for me? F is a continues function at point $ x_0 \Leftrightarrow (x_n \to x_0 \Rightarrow ...
0
votes
1answer
43 views

Proof about a subset of a metric space

Prove that a subset $A$ of metric subspace $(P, p')$ of metric space $(M, p)$ is open in subspace $(P, p')$, regarded as a metric space in its own right, if and only if there exists an open set $U$ in ...
0
votes
2answers
34 views

Fixed point in compact metric space

I guys! I try to solve the following small problem. However, I'm not able to prove the second part. In particular, I have some problems in using the compactness hypothesis on $X$ to find proper ...
0
votes
0answers
59 views

Show that $\text{int}(\mathbb Q\times \mathbb Q)$ in $\mathbb R^2$ is the empty set and find the boundary of $\mathbb Q\times \mathbb Q$.

Question: Show that $\text{int}(\mathbb Q\times \mathbb Q)$ in $\mathbb R^2$ is the empty set and find the boundary of $\mathbb Q\times \mathbb Q$. Now that I read the questions correctly (thank ...
1
vote
4answers
58 views

How to give a rigorous proof of this fact about closures of open balls in the euclidean spaces?

Let $n$ be a positive integer, $\vec{a} \in \mathbb{R}^n$, and $r > 0$. Then it is intuitively clear that the closuer of the open ball $$B(\vec{a} ; r) \colon= \{ \vec{x} \in \mathbb{R}^n \colon ...
1
vote
2answers
31 views

$C^n[a,b]$ as a normed algebra

I would like to prove that the space of complex valued functions, differentiable $n$ times with continuous derivative, $C^n[a,b]$, with the metric defined by the norm ...
2
votes
2answers
37 views

Let $(Y,\rho)$ be a metric space and $\rho : Y \times Y \rightarrow \mathbb{R}$ Prove that $\rho$ is a continuous function on $Y \times Y$.

Let $(Y,d)$ be a metric space and $d : Y \times Y \rightarrow \mathbb{R}$ Prove that $d$ is a continuous function on $Y \times Y$. I was thinking of the following : If $(a_{1},a_{2}) \in Y \times ...
3
votes
3answers
44 views

Showing $f$ is continuous on $M$ if $M=\bigcup_{n=1}^{\infty} U_n$

Let $f:(M,d)\to (N,\rho )$. If $M=\bigcup_{n=1}^{\infty} U_n$, where each $U_n$ is open, and if $f$ is continuous on each $U_n$, show that $f$ is continuous on $M$. Attempt: I note that ...
-1
votes
1answer
30 views

On Pseudometric

How a pseudometrics induces topology? Can anyone discuss on this topic or give any good reference?
0
votes
1answer
38 views

Definition of a metric space: why $E\times E\rightarrow\mathbb{R}$?

In the definition of a metric space Let $E$ be a set and $d:E\times E\rightarrow\mathbb{R}$ be a function. $d$ is a distance on $E$ if ..., why is the function $d:E\times ...
0
votes
1answer
24 views

Distance to a closed set is continuous.

I want to prove that given a metric space $(M,d)$ and $F \subset M$, then the function $f_F: M \to \Bbb R$ given by $f_F(x) = d(x,F) = \inf\{d(x,y) \ : \ y \in M\}$ is continuous. Take $x \in M$. If ...
2
votes
1answer
20 views

Extend Metric Space Challenge

Let $(E, D)$ be a metric space. Consider $D_1: E\times E \to \mathbb{R}$ where $$ D_1(x,y)=\frac{D(x,y)}{1+ D(x,y)}. $$ I read some note about it but I want to find why $D_1$ is also a metric and ...
10
votes
1answer
112 views

Is there always an equivalent metric which is not complete?

I have seen that completeness is not a topological property like compactness or connectedness. I have seen some examples also showing that there are two equivalent metrics one of which is complete and ...
0
votes
1answer
16 views

Subsets of a metric space in which Hausdorff semi-distance is symmetric

These are the definition of Hausdorff distance and Hausdorff semi-distance for subsets of a metric space $X$. ‎‎Hausdorff semi-distance of two subsets ‎$‎A‎, B‎ \subset X$ is defined as below: ‎$‎d(A ...
0
votes
2answers
25 views

Sequential Equivalence Implies Topological Equivalence

Define two metric spaces $(M,d)$ and $(M,\rho)$ to be equivalent, denoted $d\sim p$, to mean that: Topological Definition $\forall x\in M: \forall \epsilon>0 \exists \delta_1>0, \delta_2>0: ...
0
votes
2answers
24 views

What does this function converge to in $\mathbb{R}$ equipped with discrete metric?

We're given this function $f_n (x) = \begin{cases} 0 \ \mbox{ if $x <1/n$}\\ 1 \ \mbox{ if $x \geq 1/n$} \end{cases}$ I think it converges pointwise to $f(x) = \begin{cases} 0 \ \mbox{ if $x ...
2
votes
1answer
35 views

Pointwise convergence implies uniform convergence in discrete metric space $(X,d)$?

In general, does this hold for a sequence of functions in an arbitrary $X$? For a sequence to converge in the discrete metric, the sequence needs to become a constant sequence for a sufficiently large ...
0
votes
1answer
41 views

Connected Sets on Metric Spaces

I'm taking a first course in real analysis, and we're using Rudin's Principles of Mathematical Analysis as our main (only) book. In chapter two, Rudin discusses basic topology from the point of view ...
2
votes
1answer
46 views

Distance between a point and a closed set in metric space

Here is what I am thinking. Let (X,d) be a metric space and let C be a closed subset of X. Fix any poin p in X. Then, there exists a point q in C such that d(p,q) = distance(p,C). I think this ...
2
votes
1answer
31 views

What are the epis in Met?

I have an assignment to precisely describe epimorphisms and monomorphisms in Met (category whose objects are Metric spaces and whose morphisms are contractions). I have shown that Mono $\iff$ ...
1
vote
1answer
15 views

Is Cantor set $F_{\sigma}$ set or $G_{\delta}$ set?

Is Cantor set an $F_{\sigma}$ set? or a $G_{\delta}$ set? There are similar questions on stackexchange, which consider a subset of Cantor set. But, I don't find the question posted above.