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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Contraction mapping does not hold in metric space

Let $X=\mathbb{Q}\cap [1,2]$, i.e $X$ is the set of rational number between 1 and 2 inclusive. We can consider $X$ to be a metric space by endowing it with the usual distance function, i.e for $x,y ...
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102 views

Proving an inequality with $\|x\|_p$ metrics?

Let $1 \leq p < q \leq \infty$ and $x \in\mathbb{R}^n$. Show that $\|x\|_q \leq \|x\|_p \leq n^\frac{1}{p}\|x\|_q$, where $\|x\|_p$ is the metric $\left(\sum_{j=1}^n{|x_j|^p}\right)^\frac{1}{p}$. ...
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Distance between two DNA molecules $x, y$

A DNA molecule can be represented as a string of symbols $A, C, G$ and $T$, such as $GGATAATTCTAG\ldots GACCGTACCC$. For the purposes of this question, we will assume that all DNA molecules contain ...
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complete metric space

Prove or disprove: $(A_i)_{i=1}^\infty$ are closed subsets in a complete metric space. Assume that there is an open ball in the $\bigcup\limits_{i=1}^\infty A_i$ , so exists $k$ s.t $A_k$ contains ...
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Sequences in $\ell_p$ spaces

Does there exist a sequence $(x_n)$ belonging to $\ell_1\cap\ell_2$ which converges in one but not the other? $(x_n)$ is of course a sequence in these spaces, so it's a sequence of sequences.
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Euclidean distance proof

How can I show that the Euclidean distance satisfies the triangle inequality? Where the Euclidean distance is given by: $$d(p,q) = \sqrt{(p_1-q_1)^2 + \cdots + (p_n-q_n)^2}$$ Triangle Inequality: ...
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Euclidean distance vs Squared

So I understand that Euclidean distance is valid for all of properties for a metric. But why doesn't the square hold the same way?
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573 views

Taxicab Distance proof

So I am trying to prove that the taxicab distance using the triangular inequality. $$d_1(p,q)=\|p-q\|_1=\sum_{i=1}^n|p_i-q_i|$$ So I am trying to show that: $|d_1(a,b)−d_1(c,b)| \le d(a,c)$ which ...
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343 views

Long proof of equivalence of subspace and metric topology

Let $(X,d)$ be a metric space and $S\subseteq X$. Let $\tau$ be the topology on $X$ induced by $d$ and $\tau_S$ be the subspace topology on $S$: $$ \tau_S = \{S\cap V:V\in \tau\}. $$ Denote $B_r(x)= ...
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3answers
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Metric space where the distance between arbitrary points is a constant

Can I have a metric space where the distance between two points is an arbitrary constant? Does this mean that there cannot be 'co-linear' points in the space? i.e. if A B and C are colinear, and B is ...
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97 views

A question on hyperbolic geometry

I am reading a book that seems to claim the following. I suspect that there may be a misprint, or some assumptions missing. For $A=(x_1,y_1,z_1),B=(x_2,y_2,z_2)\in R^3$, define $$\langle ...
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427 views

Is a metric space perfectly normal?

I typically like to practice my knowledge on a specific concept by doing proofs using one definition of a term, and then doing the same proofs using an equivalent definition (without inducing the ...
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66 views

Is the reciprocal of the even numbers a closed subset of $(0,1)$?

This is just a quick question, as a follow-up to Chris Eagle's answer on this post. In it, he considered $X=\left \{ \frac{1}{2n} : n \in \mathbb{N} \right\}$ and $Y=\left \{ \frac{1}{2n+1} : n \in ...
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Countable complete set of limit points

Let $(X,d)$ be a metric space with $X$ - countable and such that for any $x\in X,r>0$ there exists $y\in B(x,r)$, $y\neq x$. Can $X$ be complete? I failed to prove that it cannot as well as to ...
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367 views

Is total boundedness a topological property?

If a metrizable topological space is totally bounded with one metric, is it totally bounded with all others? A related, stronger question: if every metrization of a topological space is bounded, are ...
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1answer
183 views

Does the Sorgenfrey Line have a group operation compatible with its order topology?

The title is the question, but let me explain. Let $\mathbb{L}$ denote the Sorgenfrey line. I and a friend were trying to develop some of the properties of the sorgenfrey line. (if it's metrizable, ...
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135 views

Compact Metric Spaces and Properties? [duplicate]

Possible Duplicate: Condition for family of continuous maps to be compact? I was reading through general-topology posts and I couldn't quite get this one. Here's a reformulation of the ...
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1answer
152 views

Convergence of the sequence $f_n : [0,1] \rightarrow \mathbb{R}$ defined by $f_n(x)=x^n$

From introductory real analysis, I know that the sequence of functions $f_n(x)=x^n$ converges pointwise in that $f_n \rightarrow 0$ for $0 \leq x < 1$ and $f_n \rightarrow 1$ whenever $x=1$. Thus, ...
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1answer
294 views

Metric Spaces and $\sup$ of Diameter

Question: Let $(X, d)$ be a metric space such that there is a positive $a$ and $n$ open balls $B(x_1, a),\ldots, B(x_n, a)$ such that together these balls cover $X$. Find an upper bound $M$ for the ...
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1answer
329 views

Isolated points of compact subset of the rationals

Does there exist a compact, nonempty subset of the rationals without isolated points ? My motivation was the following: If so one could define a map $f$ from the set of all compact subsets of the ...
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How Convergence in two different norms relates to Equivalence of Norms [duplicate]

Question: Why was this marked as a duplicate? The referenced question asked nearly a year later than this question. In fact I'm not even sure that they are identical at all. I was working on a ...
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5answers
997 views

Complete implies locally compact in length metric space?

I am confused. The way I see it, in a complete metric space, closed balls of finite diameter are compact since they are complete and totally bounded. Consequently a complete metric space is locally ...
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What operations is a metric closed under?

Suppose $X$ is a set with a metric $d: X \times X \rightarrow \mathbb{R}$. What "operations" on $d$ will yield a metric in return? By this I mean a wide variety of things. For example, what functions ...
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If you know the convergent sequences, how do you know the open sets?

I have a homework problem which I feel should be simple but is actually surprisingly tricky. This is why I love math sometimes.... Let $X$ be a normed linear space. Suppose $\|\cdot\|_1$ and ...
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189 views

Open ball over the real numbers

In my book, it says that any open ball $B(a,r)$ over the real numbers is equal to the open interval $(a-r,a+r)$. I wonder how I can prove that this is true, only using the metric axioms. If the ...
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156 views

Voronoi diagram with different metric functions

Given a metric space $(X,d)$ and finite number of points $(x_i)_{i=1}^n$ the Voronoi diagram (or the Dirichlet cell) $C_i$ is given by $$ C_i = \{x\in X:d(x,x_i)<\min\limits_{j\neq i}d(x,x_j)\}. ...
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131 views

Why do these inequalities in metric spaces hold?

The other day I stumbled across some inequalities regarding properties of metric spaces. I'm curious to see a proof of why it holds. Suppose $(X,\rho)$ is any metric space. For a given $\epsilon\gt ...
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145 views

What is a good way to measure the distance between finite subsets of the reals?

I have some sets of numbers, and I'd like to have a way to talk about how close these sets are to each other. I'm not sure what properties it should have (e.g. does it need to be a metric?). But I ...
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1answer
403 views

Topology: Proving a space is connected

I'm attempting to prove that a space is connected and compact. I have a continuous function $f:X \rightarrow S^{1}$. $X$ is metrizable and locally connected. $f$ is non-constant, surjective and ...
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Contraction mapping in an incomplete metric space

Let us consider a contraction mapping $f$ acting on metric space $(X,~\rho)$ ($f:X\to X$ and for any $x,y\in X:\rho(f(x),f(y))\leq k~\rho(x,y),~ 0 < k < 1$). If $X$ is complete, then there ...
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Help understanding why a complete, totally bounded metric space implies every infinite subset has a limit point

I'm reading the following proof. Properties II and III are in my title, that a complete, totally bounded metric space implies every infinite subset has a limit point. I have two questions near the ...
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How should I measure the total “closeness” of a finite number of elements?

Suppose I have n points and a way to measure the pairwise (probably non-Euclidean) distance between them. I would like to have some way to measure the total "closeness" of my points, but I'm not ...
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Chain of closed subsets in separable metric space

I need some help to prove that if $\mathcal{A}$ is a chain of closed subsets in a separable metric space then there is a countable subfamily $\mathcal{A}'\subseteq\mathcal{A}$ such that ...
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1answer
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Proper inclusion between open ball, closure of open ball and the closed ball in a metric space

In a metric space $X$ for all $x \in X, r > 0$ following is true: $B(x,r) \subseteq \overline{B(x,r)} \subseteq \overline{B}(x,r)$. Here $\overline{B(x,r)}$ is the closure of the open ball of ...
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Is $\mathbb{R}^2$ minus a countable number of points 'skew-Manhattan connected'

Let $A \subset \mathbb{R}^2$ be countable. Then it is not too hard to show that $\mathbb{R}^2 \setminus A$ is path-connected. However it is not always Manhattan connected since if $A = \mathbb{Q}^2 ...
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how can I prove formally that this set is not path connected

Let $ \mathbb{R}^2 $ with its usual topology, let $D$ the set of all the lines that pass through the origin, with rational slope. And add to $D$ some point that does not lie in any of the lines ( call ...
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Distances between closed sets on metric spaces

Which says that $\mathbb R ^ n $ the distance between a point $b$ and a set $X$ defined by $$ \inf \left\{ d\left( {b,x} \right) \,|\, x \in X \right\} $$ The proposition: If X is closed, this ...
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1answer
556 views

Completely metrizable implies $G_\delta$

It is a consequence of Lavrentyev's theorem that a metrizable space is completely metrizable if and only if it is a $G_\delta$ subset in every completely metrizable space containing it. In my ...
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Metrizable compactifications

Suppose $X$ is a metric space. When does it have a metrizable compactification? Of course it is enough to discuss complete metric spaces, but separability may not be assumed here. I know that ...
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1answer
68 views

Low distortion embeddings (reference request)

I read about the Johnson Lindenstrauss Lemma. I googled and found that low distortion embeddings is a live subject, but it seems that many interesting results are already known. Is there a book on ...
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2answers
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How can I show these two metrics give the same topology?

This question came up while showing the composition of a metric with a certain other function gives another metric. Suppose I have some metric space $(X,d)$ and a continuous, non-decreasing function ...
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1answer
225 views

Why has the extreme value function never been defined? –Or has it?

Just as when x and y are arbitrary real numbers, we often wish to consider their distance apart, and use the absolute value function to do so (namely, by means of the expression |x – y|), so also when ...
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cover a bounded set with numerable balls (sufficient conditions) metric spaces

This natural property is not true always, when I mean a bounded set, I mean a set that it's contained in some ball of radius $r$. It's not always true that if this is the case, I can cover fixing a ...
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How to calculate number of lumps of a 1D discrete point distribution?

I would like to calculate the number of lumps of a given set of points. Defining "number of lumps" as "the number of groups with points at distance 1" Supose we have a discrete 1D space in this ...
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1answer
125 views

Is it easier to make perfect sets using intersection [closed]

A perfect set $A$ is one in which every point is a limit point. So it has to be closed. Does this mean that if we want to generate perfect sets inductively it is usually best to just intersect ...
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Finite Levenshtein distance?

Is there a standard term for the relation on sequences where two sequences are related iff they have a finite Levenshtein distance, or for the equivalence classes it induces?
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627 views

A separable locally compact metric space is compact iff all of its homeomorphic metric spaces are bounded

The title is a claim my classmate made during our summer vacation :D He showed me a TeX file describing a proof of his claim, and it contains a fairly short but elegant proof. He says that the ...
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Polish space in which the interior of each compact set is empty

If anyone could give me an example of Polish space, in which the interior of each compact set is empty? I guess it is trivial, but can't find such an example.
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Something connected with Arzelà-Ascoli theorem

Let $X$ be a Polish space. Assume that $(C_m)_{m\in\mathbb{N}}$ is an increasing sequence of compact subsets of $X$ and denote $C=\bigcup_{m}C_m$. Let $\{f_n:n\in\mathbb{N}\}$ be a family of ...
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Showing the $p$-adic absolute value on $\mathbb{Q}$ is an ultrametric

Let $p$ be a prime number. Define the p-adic absolute value function $|\cdot|_{p}$ on $\mathbb{Q}$: $|x|_{p}=\left\{ \begin{array}{ll} 0 & \text{if }x=0\\ p^{-k} & \text{if ...