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 $f(x)=x\ln x$ is not uniformly continuous

Show $f:(0,\infty)\rightarrow \mathbb{R}, f(x)=x\ln x$ on $(0,\infty)$ is not uniformly continuous. I think that the general way to prove that something is not continuous in a metric space is to ...
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Show $(A^o)^c=\overline{A^c}$

Show $(A^o)^c=\overline{A^c}$. ($\rightarrow$) $(A^o)^c\subseteq\overline{A^c}$ I want to show that $(A^o)^c$ is closed and that $A^c\subseteq (A^o)^c$. Then ($\rightarrow$) follows. Since ...
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$A$ is open $\iff$ $A\cap\partial A=\emptyset$

Show $A$ is open $\iff$ $A\cap\partial A=\emptyset$. Attempt: ($\rightarrow)$ $A$ open $\implies A\cap\partial A= \emptyset$. $x\in A$ open ...
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Proving $A^o=A\setminus \partial A$

Show $A^o=A\setminus\partial A$. Attempt: I need to show inclusion on both sides, so: (a) $A^o\subseteq A\setminus\partial A$ (b) $A\setminus\partial A \subseteq A^o$ Attempt at (a): If ...
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How do I show that this metric space is not convex?

Denote $X$, the space of all sequences $\in$ $\mathbb R$. I have a metric $$d(x,y):=\sum_{n=1}^\infty 2^{-n}\frac{| x_n-y_n|}{1+| x_n-y_n|}$$ and $(X,d)$ is a metric space. How would I show that the ...
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Showing boundedness of metric space

Denote $X$, the space of all sequences $\in$ $\mathbb R$. I have a metric $$d(x,y):=\sum_{n=1}^\infty 2^{-n}\frac{| x_n-y_n|}{1+| x_n-y_n|}$$ If $(X,d)$ is a metric space and if $A$ is a subset of ...
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Continuous function on a compact metric space is uniformly continuous

I am struggling with this question: Prove or give a counterexample: If $f$ is a continuous function on a compact subset $Y$ of a metric space $X$, then $f$ is uniformly continuous on $Y$. ...
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Is the set $E$ of sequences containing only entries $0$ and $1$ in $(m,\left \| \cdot \right \|_\infty)$ complete?

I can't really wrap my head around $E$, or a Cauchy sequence in $E$. I need to take a Cauchy sequence in $E$ and show it's Cauchy in $(m,\left \| \cdot \right \|_\infty)$? I think I can show $(m,\left ...
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Distinct metrics on a manifold

I'm trying to understand basic differential geometry (my background is in mathematical logic), and I'm having a bit of difficulty with a basic point. Frequently we want to consider the set of metrics ...
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Let S be a subspace of topological space X. Show that the closure of S, the set of contact points, is indeed closed.

Let $S$ be a subspace of topological space $X$. Show that the closure of $S$, the set of contact points, is indeed closed. I need to prove that the closure is closed but I don't know how to ...
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totally bounded, complete $\implies$ compact

Show that a totally bounded complete metric space $X$ is compact. I can use the fact that sequentially compact $\Leftrightarrow$ compact. Attempt: Complete $\implies$ every Cauchy sequence ...
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$X$ compact metric space, $f:X\rightarrow\mathbb{R}$ continuous attains max/min

Let $X$ be a compact metric space, show that a continuous function $f:X\rightarrow\mathbb{R}$ attains a maximum and a minimum value on $X$. Attempt: So the important thing is that I have ...
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Proving this function is a metric

This is a space $S$ that consists of the set of all sequences of real numbers and $x=(x_1,x_2,x_3,...), y=(y_1,y_2,y_3,...)$ etc. and the metric $d$ is defined as $$d(x,y)=\sum_{i=1}^\infty ...
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If $A$ is compact and $B$ is closed, show $d(A,B)$ is achieved

Let $A, B$ be subsets of a metric space $X$. If $A$ is compact and $B$ is closed, show that the distance between $A$ and $B$ is achieved. Attempt at a proof: Let $A$ be compact and $B$ be ...
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How to prove this is a metric?

I have 2 distance functions $d(x,y)=|x^2-y^2|$ and $d(x,y)=|x^3-y^3|$ and I am trying to prove that they are metrics on $\mathbb R$, or give a counterexample that they are not metrics on $\mathbb R$. ...
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A weird subset of $\mathbb R^2$

Is there a path-connected subset of $\mathbb R^2$ such that any path connecting 2 distinct points in that subset has infinite length? I am told that there is such a set, but I don't know what it is. ...
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totally bounded metric spaces

Show that a sequentially compact metric space is totally bounded. Well since sequentially compact $\implies$ compact, any open cover of sequentially compact metric space $X$ has a finite ...
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assumed distance in metric spaces

Determine whether or not the distance between nonempty $A,B\subset X$ for metric space $X$ is assumed if A and B are closed. The definition of distance between sets A and B is ...
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compactness and boundedness

Let X be a compact metric space. Show that a continuous function $f:X\rightarrow\mathbb{R}$ is bounded. Attempt: So compact $\implies$ closed and bounded, but it is not always the case that if ...
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continuous map of metric spaces and compactness

Let $f:X\rightarrow Y$ be a continuous map of metric spaces. Show that if $A\subseteq X$ is compact, then $f(A)\subseteq Y$ is compact. I am using this theorem: If $A\subseteq X$ is sequentially ...
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999 views

General Triangle Inequality, distance from a point to a set

I am trying with no luck to prove: Let (X,d) be a metric space and A a non-empty subset of X. For x,y in X, prove that d(x,A) < d(x,y) + d(y,A)
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Metric on the unit cube

Let $X$ be $\mathbb{R}^3$ with the sup norm $\|\cdot\|_{\infty}$. Let $Y=\{x\in X: \|x\|_{\infty}=1\}$. For $x,y\in Y,y\neq -x$ define $d(x,y)$ to be the arc length of the path $$Y\cap \{\lambda ...
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If $x_n\to \overline{A}$ does it mean that $x_n\to A$?

Let $(X,d)$ be a metric space. The sequence $(x_n)$ converge to the set $A\subset X$ (denoted as $x_n\to A$) iff $$ \lim\limits_n d(x_n,A) = 0 $$ where $d(x;A) = \inf\limits_{y\in A}d(x,y)$. Let ...
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Prove that a finite union of closed sets is also closed

Let $X$ be a metric space. If $F_i \subset X$ is closed for $1 \leq i \leq n$, prove that $\bigcup_{i=1}^n F_i$ is also closed. I'm looking for a direct proof of this theorem. (I already know a ...
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triangle inequality for a certain norm

Let $d$ be a metric on a (say real) vector space $E$, with the property $$d(x,x+cy)=|c|d(x,x+y)$$ for all $x,y\in E$ and scalars $c$. I am trying to prove that $x\mapsto d(x,0)$ defines a norm. The ...
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the hyperbolic plane is complete

I'd like to prove that the hyperbolic plane is complete in a short, nice way. Here's my proof, but I'm not convinced of its correctness yet. Let $\mathbb{H}=\left\{(x,y)\in\mathbb{R}^2\big\vert ...
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For $F$ closed in a metric space $(X,d)$, is the map $d(x,F) = \inf\limits_{y \in F} d(x,y)$ continuous? [duplicate]

Possible Duplicate: Continuity of the metric function For $F$ closed in a metric space $(X,d)$, is the map $d(x,F) = \inf\limits_{y \in F} d(x,y)$ continuous? I think it is, but I'm having ...
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Why doesn't $d(x_n,x_{n+1})\rightarrow 0$ as $n\rightarrow\infty$ imply ${x_n}$ is Cauchy?

What is an example of a sequence in $\mathbb R$ with this property that is not Cauchy?
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Existence of a Riemannian metric inducing a given distance.

Let $M$ be a smooth, finite-dimensional manifold. Suppose $M$ is also a metric space, with a given distance function $d: M \times M \rightarrow \mathbb{R}_{+}$, which is compatible with the original ...
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Characterize uniform continuity by sequences/net/filter

From Wikipedia let $A$ be a subset of $\mathbb{R}^n$. A function $f : A → \mathbb{R}^m$ is uniformly continuous if and only if for every pair of sequences $x_n$ and $y_n$ such that $$ ...
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isolated points and continuous functions

Suppose $(X,d)$ and $(X',d')$ are metric spaces and $f:X\rightarrow X'$ is continuous. (a) If $A\subseteq X$ and $x_o$ is an isolated point of $A$, then $f(x_o)$ is an isolated ...
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Explanations of Lebesgue number lemma

From Planetmath: Lebesgue number lemma: For every open cover $\mathcal{U}$ of a compact metric space $X$, there exists a real number $\delta > 0$ such that every open ball in $X$ of radius ...
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Equivalence of continuity definitions

How to show that $(1)\Longleftrightarrow (2)$ in metric spaces ? pre-image of open sets are open $\delta$-$\epsilon$ definition of continuity
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Which of the following topological groups are polish or locally compact?

I want to show that the next groups are polish topological groups, which criteria should I use here? And also which are locally compact (same question)? The groups are: The group of permutations ...
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Are these equivalent definitions of the Wasserstein-metric?

Let $\rho_0$ and $\rho_1$ be two probability density functions on $\mathbb{R}^d$. The Wasserstein metric is defined as $$W_p(\rho_0,\rho_1)^p = \inf E(|X-Y|^p)$$ where the infimum is taken over all ...
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$\mathbb{Q}$ in metric space $(\mathbb{R},d)$ neither open nor closed

Show that $\mathbb{Q}\subset\mathbb{R}$ with metric space $(\mathbb{R}, d)$ is neither open nor closed in $\mathbb{R}$. Attempt: So I need to prove two parts. (a) $\mathbb{Q}$ not open: Take ...
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An open ball is an open set

Prove that for any $x_0 \in X$ and any $r>0$, the open ball $B_r(x_o)$ is open. My attempt: Let $y\in B_r(x_0)$. By definition, $d(y,x_0)<r$. I want to show there exists an ...
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intersections of sets in metric spaces

(a) Let $A_{n}=B_{1/n}((0,0))$ in $\mathbb{R}^{2}$ with the usual metric. Show that $\bigcap_{n=1}^{\infty}A_{n}$ is not open. (b) Find an infinite collection of distinct open sets in ...
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Open sets in a metric space

Prove that the following are open sets: (a) the “first quadrant,” $\{(x,y)\in\mathbb{R^2}\mid x>0 \text{ and }y>0\}$ (b) any subset of a discrete metric space I'm using the ...
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unit ball question

If $1\leq p<q$, show that the unit ball $l_{n}^p(\mathbb{R})$ is contained in the unit ball $l_{n}^q(\mathbb{R})$. Well the definition of $l_{n}^p(\mathbb{R})$ is that for ...
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Homeomorphism from square to unit circle

Can we find a homeomorphism from the square $Q_2$ of side length $2$ centered on the origin and the unit circle $S^1$? We can easily define a map $r:Q \longrightarrow S^1$ by $$(x,y) \mapsto ...
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Complete Metric Spaces

How could we show that the metric space $$X=\{0\} \cup \{\frac{1}{n}:n \in \mathbb{Z} - \{0\}\}$$ with the metric it inherits as a subset of $\mathbb{R}$ is complete? Thoughts Complete metric ...
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Does one-sided Hausdorff distance satisfy the triangle inequality for non-closed bounded subset?

Given $(X,d)$ is a metric space. Suppose that $A,B,$ and $C$ are subsets of $X$ which are bounded but non-closed. One side Hausdorff distance is defined by $$d(A,B)= \sup_{x\in A} \inf_{y \in B} ...
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A metric on $\mathbb{C^n}$ [duplicate]

Possible Duplicate: Show that $d$ is a metric on $\mathbb{C^n}$ On $\mathbb{C^n}$, define $||z||=(\sum_{j=1}^{n}|z_{j}|^{2})^{1/2}$ and for $z,w\in\mathbb{C^n}$ define $d(z,w)=||z-w||$. ...
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Cosine similarity / distance and triangle equation

There is a similarity function particular popular for processing sparse vectors such as textual data (word frequency counts etc.) commonly referred to as cosine similarity. There are two variants to ...
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Topology induced by metric space

This a problem from Topology - James Munkres. Let $X$ be a metric space with metric $d$. Let $X'$ denote a space having the same underlying set as $X$. Show that if $ d: X'\times X' ...
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Show that $d$ is a metric on $\mathbb C^n$

On $\mathbb C^n$, define $||z||=(\sum_{j=1}^{n}|z_j|^2)^\frac{1}{2}$ and for $x,z\in\mathbb C^n$ define $d(z,w)=||z-w||.$ Prove that $d$ is a metric on $\mathbb C^n$. My attempt: I need to show ...
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$Tx = \frac{x}{2} +\frac{1}{x}$ is a contraction on $M = [1,\infty)$ in $(\mathbb{R},|\cdot|)$?

I cannot seem to find a contraction factor such that $$Tx = \frac{x}{2}+\frac{1}{x}$$ is a contraction on the whole set $[1,\infty)$ in the complete normed space $(\mathbb{R}, |\cdot|)$. My argument ...
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Embedding tree metric isometrically into $\ell_\infty$

I just started (independent) learning on metric embeddings from the Fall 2003 offering of the course at CMU. I have a limited mathematical background and alas, it made me stumble at the first exercise ...
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Total boundedness, an equivalent expression

I'm trying to show that a metric space $(X,d)$ is totally bounded iff every sequence in $X$ has a Cauchy sub-sequence. This is a point that rose up the other day in another topic and it seemed like ...