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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Showing that a set is closed

Show that the set $S=\{a \in \mathbb{R}^3\,| \,a_1 +a_3^2 \sin(a_1+a_2)\geqslant a_3\}$ in closed in $\mathbb{R}^3$ with the euclidean metric. I know that I would probably have to show that the ...
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Growth of Distance Function

Suppose I have a distance function $$D(P,Q) = x$$ where $P$ and $Q$ are multivariate normal distributions and $x \in \mathbb{R}$. I want to study the behavior of the function as the dimension of ...
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60 views

Infinite intersection and limits

I'm having difficulty understanding the relationship between a limit and an infinite intersection. Any help would be greatly appreciated. Specifically, suppose we take any non-increasing sequence of ...
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30 views

Extract a converging geodesic from a sequence

Let $(X, d)$ be a compact, complete, separable metric space, and $g_n$ a sequence of constant speed geodesic with the same endpoints, i.e. continuous maps $g_n : [0,1] \rightarrow X$ such that $$ ...
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Continuity of metric

I recently came across this definition: Let $(X,d)$ be a metric space and $A$ be a nonempty subset of $X$. For each $x\in X$ we define a distance from $x$ to $A$ by the equation $d(x,A)=inf\{d(x,a) | ...
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35 views

Existence of a geodesic in a complete separable metric space

If I have $X$ a complete separable metric space, $x, y \in X$ arbitrary points, how can I define a constant speed geodesic, i.e. a continuous map $g : [0,1] \rightarrow X$ such that $$ d(g(t), g(s)) = ...
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32 views

Compact space with $x_{n_j} \to x $ for all conv. subsequences

Given a compact metric space $(X,d)$ with sequence $(x_n)_n \subseteq X$ and every convergent subsequence of $(x_n)_n$ converges against $x$. How can I show that $x_n \to x$? Hints are welcome! ...
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152 views

Show that $f:[0,1] \to [0,1]$ is continuous if $f(x) = x^{1/k}$ for any $k \in \mathbb N$

I'm very confused right now and I want to apply the theorem that says " A mapping f of a metric space $X$ into a metric space $Y$ is continuous on $X$ if and only if $f^{-1}(V)$ is open in $X$ for ...
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142 views

Showing that $(S,d)$ is a metric space for $S = (0,1]$ and $d(x,y) = |1/x - 1/y|$

Let $S$ be a half-open interval $(0,1]$. If we define $d$ on $S$ by $$d(x,y) := \left|{1\over x} - {1\over y}\right|\;,$$ then show that $d$ is a metric on $S$. Also, prove that $(S,d)$ is a ...
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153 views

Show that the following is a metric on $\mathbb R$

Is the following just a matter of showing the 3 properties that make up a metric?? Define d on $\Bbb R\times\Bbb R$ by $d(x,y)=\min \{1,|x-y|\}$. Show that $d$ is a metric on $\Bbb R$ $d(x,y)=0$ ...
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74 views

how to prove this is a metric given the following conditions

I need help wrapping my head around the concepts of metrics and how to prove that something is a metric. For example, prove that if $p_1$ and $p_2$ are metrics in $X$, then $p_1 + p_2$ and $\max\{p_1, ...
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56 views

How to find the closure of a subset

How do I start find the closure of a subset? Let's say I'm given a list, such as $$A=\left\{\frac12,\frac13,\frac23,\frac14,\frac24,\frac34,\frac15,\frac25,\frac35,\frac45,\cdots\right\}$$ using the ...
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1answer
135 views

Homeomorphism of closed intervals

One can prove that if $f: [a,b] \to [f(a), f(b)]$ is continuous and monotone increasing that then it is a homeomorphism. The only part one might have to think about at all is that $f$ is open but that ...
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330 views

Sequence has a convergent subsequence in R^n

Suppose A is a closed and bounded subset of R^n. Let {ak} be a sequence in A. Thus, the elements of {ak} are: (a11,a12,...,a1n), (a21,a22,...,a2n), ... ... (ak1,ak2,...,akn), ... We are not sure if ...
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284 views

Existence of limit for convergence by measure for Cauchy-in-measure sequence+completeness of metric space?

Sorry if this is the wrong place to put it. But this question come from a graduate level textbook and seems pretty hard to me, so I hope this is a good place. Anyway, this come from the book Real ...
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Finding an isometry given the distance between points

If $A,B,C,D \in \mathbb{R}^n$, such that $d(A,B)=d(C,D)$, then exists an isometry $f:\mathbb{R}^n\rightarrow \mathbb{R}^n $, such that $f(A)=C, f(B)=D$. Thanks a lot for the help.
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265 views

Proving subset is not connected iff there exist open sets in X

Prove that E $\subseteq$ X is not connected if and only if there exist open sets $A, B \subseteq X$ such that $E \subseteq A ∪ B, A ∩ B$ = $\emptyset$ and $E ∩ A$ and $E ∩ B$ are both nonempty. $X$ ...
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64 views

Flat space Minkowski metric

I am having some problem understanding the why in Minkowski spacetime, the continuity equation is written as $$\partial_\mu J^\mu=0.....................(*)$$ Physically, I know that $$\partial_t ...
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47 views

Proving sequential compactness from open cover compactness.

Let $(\mathcal M,d)$ be a metric space and $A\subset\mathcal M$. The following types of compactness are equivalent: (i) Each open cover of $A$ contains a finite subcover. (ii) $A$ is sequentially ...
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270 views

Differentiation/Integration continuous function (metric spaces)

I have the following two questions: Is differentiation, $f(x) \mapsto f'(x)$ a continuous function from $C^1[a,b] \longrightarrow C[a,b]$ and Is integration, $f(x) \mapsto \int_a^x ...
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Idempotent of closure and other properties

I have to prove: $\mbox{ClCl}A = \mbox{Cl}A$ $\mbox{diamCl}A = \mbox{diam}A$ If $G$ is open set then for any $A$ we have $ \mbox{Cl}(G \cap A) = \mbox{Cl}(G \cap \mbox{Cl}A)$ My attempt: ...
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663 views

Prove that if a function $f: X\to Y$ continuous then its graph is closed

The graph of $f$ is $G(f) = \{(x,f(x)) : x\in X\} \subseteq X\times Y$ $X$ and $Y$ are metric spaces. a) Suppose $f$ is continuous and prove that $G(f)$ is a closed set. b) Suppose that $G(f)$ is ...
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133 views

$C^1[a,b]$ is closed in $C[a,b]$

So, I have the following question: Let $C[a,b]$ denote the space of continuous real-valued functions on $[a,b]$ with the sup-metric. Let $C^1[a,b]$ denote the space of continuously ...
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46 views

what is the limit of $l_p$ at p=0?

The p-norm is defined as: $$ \ \|x\|_p=\left(|x_1|^p+|x_2|^p+\dotsb+|x_n|^p\right)^{\frac{1}{p}} $$ When $p<1$, this is no longer a "norm" because it violates the triangle inequality (- it is ...
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187 views

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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66 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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26 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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39 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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53 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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103 views

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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135 views

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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158 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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124 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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30 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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1answer
174 views

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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53 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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49 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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57 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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Is the space of converging sequences in N with metric space d(n,m) = |n-m| countable? [duplicate]

I have no clue in this problem.... Thanks everyone in advance.
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70 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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37 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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1answer
49 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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45 views

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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112 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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40 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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1answer
137 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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32 views

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 ...