Tagged Questions

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Question about Boundary points of the sets in metric space

Let A be a metric spaces. Prove the following properties: The boundary of $A$ equals $A'-A$ The boundary of $A$ is the closed set. $A$ is closed if and only if it contains its boundary. Where ...
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Number of open sets in a metric space

I have got the following question which I could not solve: can a metric space have exactly 36 open sets? I believe if the metric space is finie, then it has to be discrete and so the number of open ...
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For compact $K$ and open $U \supseteq K$, there exists $\varepsilon>0$ such that $B(K,\varepsilon) \subseteq U$

Let $X$ be a metric space. Let $K$ be a compact subset of $X$ and $U$ an open subset of $X$ containing $K$. I strongly believe and want to prove that there exists $\varepsilon>0$ such that ...
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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 ...
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Subset of infinite connected set

How to proove that infinite connected set has got proper infinite connected subset?
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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 $d_1(x, y) \le C\, d_2(x, y)\ \ \ \forall x, y \in X$. ...
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Compactness and sequential compactness in metric spaces

I got a question: I'm trying to proof that every metric space is compact if and only if the space is sequentially compact. In all the proves I have found, they used the Bolzano-Weierstrass theorem. Is ...
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Compact set in $(\mathbb R,\rho_1)$

$P = \mathbb R, \rho(x,y):|x|+|y|$ if $x \ne y$ or $0$ if $x=y$. Question: is $[-1,1]$ in $(P,\rho)$ compact set? I think yes: $[-1,1]$ is bound set, all sequences in it also bound, and by ...
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Continuous function in metric spaces

$P = \mathbb R, \rho(x,y):|x|+|y|$ if $x \ne y$ or $0$ if $x=y$. $f(x): (P,\rho) → (\mathbb R,\rho_1): 0$ if $x∈[-1,1]$ or $1$ if $x∈\mathbb R/[-1,1]$. $\rho_1 (x,y) = \sum_{k=1}^\infty |x_k-y_k|$. ...
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Let $M$ be a bounded subset of the space $C_{[a,b]}$. Prove that the set of all functions $F(x)=\int^{x}_{a}f(t)dt$ with $f\in{M}$ compact.

Let $M$ be a bounded subset of the space $C_{[a,b]}$. Prove that the set of all functions $F(x)=\int^{x}_{a}f(t)dt$ with $f\in{M}$ compact. Some helpful definitions: bounded - A subset $S$ of a ...
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A complex metric

Given the following definition $d(z , w) = \begin{cases}0 & z=w \\ |z|+ |w| & z\neq w \end{cases}$ I have to prove that $d(z,w)= 0\Rightarrow z = w$ Which is in part of checking that $d$ is ...
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Show the following is Cauchy:

I am trying to prove that the Euclidean Norm/inner product on C([0,1]) does not give rise to a complete metric space. To do this I am trying to find a Cauchy Sequence which does not converge in ...
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(i) Show that T is continuous on $(X,d)$. (ii) Show that T is continuous on $(X,d_{2})$.

Let $K(t,s)$ be a continuous function on $[0,1]\times{[0,1]}$. Let $X=C[0,1]$ be the set of continuous functions defined on the interval $[0,1]$. Define the mapping $T:X\rightarrow{X}$ by: for every ...
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Conditions to make a function a metric on $\mathbb{R}$?

Let $f:\mathbb{R}\rightarrow \mathbb{R}$. What conditions ensure that $d(x,y)=|f(x)-f(y)|$ defines a metric on $\mathbb{R}$ Let $g:[0,\infty) \to \mathbb{R}$. What conditions on $g$ ensure that ...
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Is this claim harder to prove for arbitrary metric spaces than for the reals?

Let $X$ and $Y$ be metric spaces. Define the distance between functions $f, g$ from $X$ to $Y$ as $$d(f, g) = \sup_{x \in X} \frac{d(f(x), g(x))}{1+d(f(x), g(x))}$$ Is it true that if $f_n:X \to Y$ ...
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Prove that $E$ is disconnected iff there exists two open disjoint sets $A$,$B$ in $X$

Let $(X,d)$ be a metric space. Prove that $E$ is disconnected iff there exists two open disjoint sets $A$,$B$ in $X$ such that $E\cap A\neq\emptyset, E\cap B\neq \emptyset$ and $E\subset A\cup B$. ...
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Prove that $\exists\ C$ such that $T: C \to C$ with $C=\overline{\text{conv}}T(C)$

I have a problem: Let $K$ be a nonempty, closed, bounded and convex subset of reflective Banach space $X$. Suppose that $$T:K \to K$$ is nonexpansive. Prove that $\exists\ C$ such that ...
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Prove that every pseudocompact metric space is compact

This is from Real Mathematical Analysis by Pugh, problem 2.85(a). I've seen proofs but they've used concepts that haven't been covered up to this point, like the Tietze extension theorem, metrizable ...
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Open sets are $\mu*$-measurable in a metric space with a given condition

I have a homework problem that I'm very stuck on. The problem statement is as follows: "Suppose that $X$ is a metric space, and that for any sets $E,F \subseteq X$, if dist$(E,F) > 0$ then ...
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