Metric spaces are sets on which you can measure the "distance" between any two points. The distance measurement is generally required to be symmetric (so distance from $A$ to $B$ is the same as distance from $B$ to $A$), positive for two distinct points, and obeying the triangle inequality.

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Show that the Cantor set is perfect i.e. $C=C'$

Let $C\subseteq \mathbb{R}$ be the classical Cantor set, show that $C=C'$. So this is what i've done so far: Take $x\in C$ $ \Rightarrow x \in C_{k}\text{ }\forall k$ $\Rightarrow \forall k ...
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Complete Condition in Banach Fixed Point Theorem

Can someone provide an example to show that for the Banach fixed point theorem, that is if $T : X → X$ is a contraction in a complete metric space $(X, d)$ then $T$ has a unique fixed point that $X$ ...
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110 views

Infinite-dimensional normed spaces and the distance

By $X$ we denote an infinite-dimensional normed space (it seems to be obvious that the case of finite dimension is not suitable). Let $X_0$ be a closed subspace of $X$ and $x\in X$. Then there is the ...
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33 views

Check the convergence of a sequence

Let $X$ =$[0,1]$ and $d(x,y)=|x-y|/(1+|x-y|)$ be the metric defined on $X$. Then check whether the sequence ${x_n = 1/n^2}$ A) Converges in $(X,d)$ B)Does not converge in $(X,d)$ My attempt : I ...
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42 views

Calculus continuity question.

show that the function f(x,y)= |x-1| + |y-1| is continuous at (2,2) using epsilon delta definition. The way I have done this is as follows. |f(x,y)-f(2,2) = ||x-1|+|y-1|-(1+1)| = ||x-1|+|y-1|-2| ...
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61 views

Check whether the following is a metric

I got this question on an internal today, Check whether $e(x,y)$ = $d(f(x),f(y))$ for any function $f:X \rightarrow X$ is a metric on $(X,d)$. I think that I have messed it up. My argument was ...
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Completeness of metric space induced by outer measure (similar to Nikodym metric)

Let $S_\mu$ be a semi-ring of subsets of $X$ and $\mu$ be a $\sigma$-additive measure on $S_\mu$. Let $\mu^*$ be the induced outer measure on $P(X)$. Define a relation $\sim$ on $P(X)$ by ...
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How can one prove that mahalanobis distance is a metric?

How can one prove that mahalanobis distance is a metric? How can one show that these four properties of a metric are valid for mahalanobis distance? 1) d(x, y) ≥ 0 (non-negativity, or separation ...
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324 views

Restrict a metric, gives same topology as subspace topology from larger space $X$

Let $A\subseteq X$. If $d$ is a metric for the topology of $X$, show that $d\restriction_{A\times A}$ is a metric for the subspace topology on $A$. I've shown that $d'=d\restriction_{A\times A}$ ...
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442 views

Metric is continuous, on the right track?

Let $X$ be a metric space with metric $d$. Show that $d:X\times X\rightarrow \mathbb{R}$ is continuous. The problem is taken from Munkres Topology second edition, Section 20. I know that if $d$ ...
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34 views

Why is no non empty open suset of $\mathbb{R}$ open in $\mathbb{C}$?

I was studying about topologies of metric subspaces and superspaces. I came across this example: Every open subset of $\mathbb{R}$ is the intersection of $\mathbb{R}$ with an open subset of ...
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Proving completeness of Nikodym Metric

I'm trying to prove completeness directly of the metric given by $d(A, B) = \mu (A \triangle B)$ on a finite measure space $(X, M, \mu)$. Edit: I should make clear that I'm referring to completeness ...
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58 views

Why is this set closed?

Let $(X,d)$ be a metric space. Let $a \in X$ and $r \ge 0$. Define: $E_r(a) = \{b \in X : d(a,b) \le r\}$ I want to show that $E_r(a)$ is closed. Here's what I know: $E_r(a)$ is closed if every ...
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71 views

Metric Space- open sets

$\qquad\mathit{(i)}\,$ We know that $\sin:\Bbb R\to\Bbb R$ is continuous. Show that, if $\,U=\Bbb R$, then $U$ is open, but $\sin U$ is not. $\qquad\mathit{(ii)}\,$ We define a function $f:\Bbb ...
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138 views

Boundary of $M_rp$ not equal to the sphere of radius $r$ at $p$?

My problem is to find a metric space in which the boundary of $M_rp$, where $M_rp = \{q \in M: d(p, q) < r \}$, is not equal to the sphere of radius $r$ at $p, \{x \in M: d(x, p) = r\}$. ...
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104 views

Prove d to be a metric

Goodday. The problem is as follows: Let $\mathbb{Z}^\mathbb{N}:=\{x:\mathbb{N}\rightarrow \mathbb{Z} \}$. We define a function $\text{d}:\mathbb{Z}^\mathbb{N} \times \mathbb{Z}^\mathbb{N} ...
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Is Minkowski space not a metric space?

I've just started reading a book on functional analysis, and first definition given there is for a metric and metric space: Let $\mathfrak{M}$ be an arbitrary set. A function $\rho\colon \mathfrak ...
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43 views

Reparametrization of an absolutely continuous curve

If $\alpha : [0,1] \rightarrow \mathbb{R^n} $ is $C^1$ and $\alpha'(t) \neq 0$ for all $t\in[0,1]$ then there always exists a reparametrization in which $\| \alpha'(s) \| = 1$. Is there an equivalent ...
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80 views

What do point functions and point like functions actually mean?

What do these actually mean ? I know the mathematical definition but i don't think that i truly understand there true meaning. Point functions: Suppose $(X,d)$ is a metric space and $z \in X$. Then ...
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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 ...
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Either show $c_0=\{x=(x_1,x_2,x_3,…) | x_i\in \mathbb{C}, x_i \rightarrow 0\}$ is complete or closed under supremum norm.

Let $l^\infty =\{x=(x_1,x_2,x_3,...)|x_i \in \mathbb{C}, \|x\|_\infty=\sup_{i\in \mathbb{N}}{|x_i|}<\infty\}$. and $c_0=\{x=(x_1,x_2,x_3,...) | x_i\in \mathbb{C}, x_i \rightarrow 0\}$. So ...
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170 views

Every limit point is the limit of a sequence

Assume we have a metric space $X$, a subset $E\subseteq X$, and a limit point $p$ of $E$. Proofwiki and Rudin both "construct" a sequence that converges to $p$ using the fact that every neighborhood ...
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45 views

neighborhood space metric

Let $M$ be a metric space and $a \in M$. We say that $V \subseteq M$ is a neighborhood of $a$ when $a \in \operatorname{Int}(V)$. Show that if $(x_n)$ is a sequence in $M$, then the following are ...
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space metris disjoint

Let F be a closed subset of a metric space M and p∈M∖F . Show that there are two disjoint open sets G and H in M such that p∈G and H⊆F . I solved well but I think is not the way ... We take a ...
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209 views

Proving that a holder continuous function always has a smaller exponent.

According to wikipedia if we have $f:X \rightarrow Y$ which is $\alpha$-Holderian then for all $\beta < \alpha$ the function is also $\beta$-Holderian. How do we prove this starting from the fact ...
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space metric converges

Hi my name is Viviane'm Brazilian and I need urgent help with these questions: A) Use the definition to find the limit of the sequence $x_n = \frac {n}{ (n + \sqrt 3)}$ $\Bbb R$-converges with the ...
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54 views

prove that metric and series

Denote $E$ the set of all real sequences $\{a_n\}$ such that $|a_n| \leq 1$ for every positive integers $n$.Let $\{a_n\},\{b_n\} \in E$ Prove that ...
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37 views

Closure Criterion for convergence of sequences

I know that $\{z\}=\bigcap\{\operatorname{cl}\,\{x_n\mid n\in S\} \mid S\subseteq \mathbb{N}\ \text{and}\ S\ \text{is infinite}\}$ is one of the criteria's of convergence of sequences in a metric ...
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Connectedness of a union of sets.

Assume that $E$ and $F$ are conneceted subsets of the metrix space X, such that $\bar{E} \cap F \neq \emptyset$. Prove that $E \cup F$ is conneceted as well. When I draw A picture the statement ...
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Question on dense sets

We have a function $f(x)$, $x \in X$ where $X$ is a complete metric space and say $f()$ is continuous. Then say $f(y)=g(y)$, $y\in Y$, $Y\subset X$, $g()$ is continuous in $Y$, and $Y$ is dense. (a) ...
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Inclusions to complete spaces

Let $X$ be any bounded metric space and $B(X,\mathbb{R})$ - a set of all bounded functions $X\rightarrow \mathbb{R}$ which we endow with a norm $||f||=\sup_{t\in X}|f(t)|$. It is easy to verify that ...
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$A \subseteq (X,d)$ 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 ...
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Is the set closed, open, or neither?

Consider $C[0,1]$, the normed linear space of all real-valued continuous functions within the given interval. The norm endowed on this space is $\|f\|_{\infty} = \sup_{x \in [0,1]} |f(x)|$. Consider ...
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Is a set of single element $\{x\}$ connected in a metric space $(X,d)$?

Is a set of single element $\{x\}$ connected in a metric space $(X,d)$? Definition: Suppose that $(X,d)$ is a metric space. A set $E \subseteq X$ is said to be disconnected if there exist two ...
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Is there a nonstandard characterization of Lipschitz continuity?

Let $f: \mathbb R \to \mathbb R$ be Lipschitz continuous with finite constant $L$. Then $$ |f(x) - f(y) \le L |x-y|, \tag{1} $$ and, by direct transfer, this property holds for $^*\!f$. For ...
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76 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 ...
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Proving it doesn't exist a homeomorphism between $\mathbb R$ and $\mathbb R^n$, $n>1$.

I have to prove that for $n \ge 2$, there doesn't exist a homeomorphism between $\mathbb R$ and $\mathbb R^n$. Could anyone give me a hint on how could I prove this?
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Show that a metric space has uncountably many disjoint dense subsets if every ball is uncountable.

I was wondering how to show that a metric space $X$ has uncountably many dense subsets $Y_{\alpha}$ such that $Y_{\alpha_1} \cap Y_{\alpha_2} = \emptyset$ if $\alpha_1 \neq \alpha_2$, under the ...
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Show that dist(A,B) is a metric

this is my first post and wasn't able to find this question anywhere. I am trying to show, that $$\text{dist}(A,B):=\text{inf}\left \{ d(a,b): a\in A, b\in B\right \}$$ is a metric on $\mathcal{P}(X) ...
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Help with a proof regarding empty interior

Show that every countable subset of $\mathbb{R}$ has empty interior in $\mathbb{R}$ and therefore is included in its own boundary ? I have no idea how to proceed with this one. Any help would be ...
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closure and interior of subsets of a metric space

Suppose $X$ is a metric space and $S$ and $A$ are subsets of $X$. If $S \subset A \subset Cl(S)$ , then $Cl(A) = Cl(S)$. Also if $Int(S) \subset A \subset S$, then $Int(A) =Int(S)$. What if, ...
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Why is the diam ($Q$) infinite?

I was trying to find a counterexample to show that the $diam(A)$ and the $diam(Int(A))$ may not be the same, where $A$ is the subset of the metric space $X$. I chose the metric space $X$ = $R$ , and ...
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topology - triangle inequality

In my homework, I have this question: Given points $u,v,w,x$ in a metric space $(X,d)$, prove that $$|d(u,v)-d(w,x)|≤d(u,w)+d(v,x).$$ Use this result to prove that: For sequences ...
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68 views

Do $L^P$ functions form a metric space?

I have a general question about $L^P$ functions. I have heard that $L^P$ functions form a vector space. My question is can we make them form a metric space too? And what is/are the possible metric/s ...
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Removing isolated points to get a perfect set

The motivating question is the following: If $F$ is a closed subset of $\mathbb{R}^1$, can one find a perfect set $E\subset F$ such that $m(E)=m(F)$ (in Lebesgue measure)? Define $F_0=F$ and ...
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28 views

Checking of continuity

While compactness & connectedness are preserved under continuous maps, this question comes to my mind: $f : \mathbb R \to \mathbb R$ is strictly monotone increasing function such that {$ f(x) : x ...
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Question regarding diameter of subsets of a metric space

The question is : Find a condition on a metric space$(X,d)$ that ensures that there exist subsets $A$ and $B$ of $X$ with $A \subset B$ such that $diam(A)$ = $diam(B)$. I know that if $X$ is a ...
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86 views

Is this subset of $\mathbb{R}^2$ closed?

I have the set $S = \{(x,y) | x>0, y\geq0\}$ and am asked if it is open/closed. I have proven that it's not open but I'm confused about whether it's closed, as its complement is not open. Am I ...
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topology homework

I'm new to topology, and therefore not very good at it yet. I have following questions, that I have ansewer, please help me verify what is not correct and what is missing in my answers. Let $X$ be ...
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1answer
45 views

Definition of accumulation points

Suppose $X$ is a metric space, $z \in X$ and $S \subset X$. Then $z$ is called an accumulation point of $S$ in $X$ if, and only if, dist(z,S\z) = 0. But such points need not be members of $S$, ...