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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Creating a configuration of points where each point is away from all other points by a pre-defined distance

Let's assume that the points $\in \mathbb{R}^2$ and there are only C=5 points (in practice, I may have $\mathbb{R}^{800}$ and 1000 points). The first out of the five points is fixed. We also have been ...
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Topology - Open set [closed]

How do I show this? Let $(X,d)$ be a metric space and $x\in X$ an element in $X$. Show that \begin{align*} \{y\in X|d(y,x)>r\} \end{align*} is open for any $r\in \mathbb{R}$. The definition of a ...
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50 views

Is it always possible to find a “pre-metric” from a metric?

Problem Let $X$ be a non-empty set. Let $f:X\times X\to \mathbb{R}$ satisfying the following properties, $f(x,y)=0\iff x=y$ for all $x,y\in X$. $f(x,y)=-f(y,x)$ for all $x,y\in X$. ...
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Are the set of probability functions with compact support in a fixed closed ball complete under the Wasserstein norm?

Let $B_R$ be a closed ball of radius $R$ in the space $\mathbb{R}^d$. As the title suggests I have this feeling that the set of functions $$S:= \left\lbrace f:\mathbb{R}^d \to \mathbb{R} ...
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Theorem 2.41 in Baby Rudin: Is this proof good enough? Can we generalise it?

Here is Theorem 2.41 in the book Principles of Mathematical Analysis by Walter Rudin, 3rd edition: If a set $E$ in $\mathbb{R}^k$ has one of the following three properties, then it has the other ...
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18 views

Are sequences themselves metric spaces with the inherited metric?

I have been asked to show whether a sequence $(p_n)$ in $\mathbb{R}$ is a metric space with the inherited metric $d(x,y) = |x-y|$ It seemed to me at first to be a slightly odd question because we ...
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2answers
57 views

If two nested open sets have the same nonempty boundary, are they the same set?

Let $(X,d)$ be a metric space. Let $B_\epsilon(x)$ be the open ball of radius $\epsilon$ centered at $x$. For $x\in X$ and $\epsilon>0$, suppose that $V$ is an open set in $X$ with $V\subseteq ...
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Subset of $(l^{2},d_{2})$ is open

Show that $A = \{\phantom{i}\{x_{n}\} \in l^{2} \hspace{2mm}:\hspace{2mm} |x_{n}| < 1, \forall \phantom{i}n \in \mathbb{N}\phantom{i} \}$ is open in $(l^{2},d_{2})$. The $d_{2}$ metric is: $$ ...
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Theorem 2.27 (a) in Baby Rudin: Is his proof complete enough?

Here's Theorem 2.27 (a) in the book Principles of Mathematical Analysis by Walter Rudin, 3rd edition: If $X$ is a metric space and $E \subset X$, then $\overline{E}$ is closed. Now here's ...
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1answer
37 views

Continuous functions with values in separable Banach space dense in $L^{2}$?

Let $[0,1] \subset \mathbb R$ be a the compact interval in the real numbers $\mathbb R$. We know that $C([0,1] \to \mathbb R)$ (the continuous function on $[0,1]$ with values in $\mathbb R$) are dense ...
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Why is it that $\inf_{x∈E} d(x, a) > 0$?

Suppose $X$ is a metric space, $E ⊂ X$ is closed, and $a$ is a point not in $E$. Why is it that $$\inf_{x∈E} d(x, a) > 0$$ ?
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38 views

Equivalent conditions for continuity in a metric space

Let $f: (X,d_{X}) \to (Y,d_{Y})$. We have that $f$ is continuous iff for any open subset $U \subset Y$, $f^{-1}(U)$ is open in $X$. Prove the following are also equivalent to the definition for the ...
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2answers
60 views

Elementary geometric characterization of spheres?

I've read the following two theorems. Theorem. A compact connected metric space whose points are cuts points with the exception of at most two is homeomorphic to the unit interval. Theorem. A ...
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24 views

Limit Points of closure of A is subset of limit points of A

For a metric space $(M,d)$ and a subset $A \subset M$, is it true that the set of limit points of the closure of $A$ is a subset of the limit points of $A$? ( I have managed to prove the reverse ...
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2answers
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If $|X|<\infty$ then T metrisable $\rightarrow$ T discrete topology.

If $|X|<\infty$ then T metrisable $\rightarrow$ T discrete topology. I said let $d$ be a metric and let $x \in X$. I want to show that $\{x\}$ is open. How do I show this?
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18 views

How can we find a contradiction?

Let $(X, \rho)$ be a metric space and $x \in X, A \subset X (A \neq \varnothing)$. We have $x \in \overline{A}$ iff $d(x,A)=0$. We suppose that $d(x,A)=0$ . We want to show that $x \in ...
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37 views

how do I prove the following set is open?

let $(X,d)$ be a metric space and let $A$ and $B$ be two disjoint closed sets in $(X,d)$. define $U=\{x\in X | d(x,A)<d(x,B)\}$. how do I prove that $U$ is open in $(X,d)$?
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36 views

Show a unit disk with max metric is closed?

I must show that if $\mathbb R$2 is equipped with max metric, d = (($x$1, $x$2),($y$1, $y$2)) = max{|$x$1 - $y$1| , |$x$2 - $y$2|} then the disk D = {($x$1, $x$2) ∈ $\mathbb R$2 : $x$12 + $x$22 ...
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1answer
24 views

Interior points: Precise definition

Let $(M,d)$ be a metric space. Then $ x \in S $ is an interior point of $ S $ if some ball centered around S of positive radius is wholly contained in $S$. But consider this. The set $S_{L}$ of all ...
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1answer
40 views

Why is $U ⊂ \mathbb{R}^n$ open with respect to metric $d_p$ iff it is open with respect to metric $d_q$ fo $q ∈ [1, ∞)$?

Let's say that for any $p ∈ [1, ∞)$ we have a distance function on $\mathbb{R}^n$ given by $$d_p(x, y) := \left(\sum^n_{j=1}|x_i - y_i|^p\right)^{\frac{1}{p}}$$ How would I show that a set $U ⊂ ...
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Single reference to classical results in analysis.

I am writing an expository work. And I need classical references (books or articles) that simultaneously proof the three classical results below. Any suggestion? Theorem. Let ...
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Name for metric with property that each point is separated…

Let $(X,d)$ be a metric space such that for each point $x \in X$ we have the property that $\inf\{d(x,y) \,\vert\,x\neq y,\,y \in X\} > 0$. i.e each point can be separated by some small ball. Is ...
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27 views

Use sequences (and limit points) to show a set is closed?

I was working on a problem and wanted to use limit points and their sequences to show that a set is open (by showing it's complement is closed). I got through the whole thing only to realize I didn't ...
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41 views

Distinct topologies on continuous functions $(L_p$ spaces restricted to $C[0,1])$

I am trying to show that the topologies $\tau_p$, $\tau_q$, $\tau_{\infty}$ ($1\leq p<q$) induced by the metrics $$\delta_p(u,v)=\left(\int_0^1 |u-v|^p\right)^{1/p}\quad ...
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+300

How many metrics are there on a set up to topological equivalence?

I want to find the number of topologically nonequivalent metrics on a set. I think if the cardinal of set is finite then we have one metric that is the discrete metric and every metric on this set ...
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19 views

Space of real sequences is separable

My task is to prove that the metric space $(\textbf{s},d_{\textbf{s}})$ is separable, where $\textbf{s}$ is the set of any real sequence with metric defined by $$d_{\textbf{s}}(x,y) = ...
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Subspace of a separable space is separable

Let $(X,d)$ be a separable space and $Y \subset X$. Show that $(Y,d)$ is also separable. My approach is as follows: Let $(X,d)$ be a separable space and $Y \subset X$. Since $X$ is separable, ...
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Showing that the metric $d$ is a norm

Let $X$ be a vector space, and $d:X\times X \to \mathbb{R}$ is a metric on $X$. Also suppose that $d$ is invariant under translations, i.e. $d(x,y)=d(x+z,y+z)$ for all $x,y,z \in X$. Is $d(x,y)$ for ...
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If A is nowhere dense in M, and if G is a nonempty open set in M, prove that A is nowhere dense in G.

If A is nowhere dense in M, and if G is a nonempty open set in M, prove that A is nowhere dense in G. I tried by contradiction but could not figure it out. I found that we can use following result ...
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1answer
46 views

uncountable co-meagre set in Polish Spaces

Let $X$ be an uncountable Polish Space and let $Y$ be a co-meagre subset of $X$. How can I prove that $Y$ is uncountable? Possibly proof without using borel sets. Thank you
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1answer
57 views

Prove that the following statements are equivalent

Let $f: (X,d) \rightarrow (Y, d')$ be a function. Prove that the following are equivalent: $f$ is continuous . For every $A \subset X$, $f(cl(A)) \subset cl(f(A))$. For every closed set $B$ in ...
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32 views

Closed sets and how they relate to open balls?

Is it correct to say: If a set $A$ has a point $x$ such that for all $r>0$, the open ball of radius $r$, centered at $x$ is not a subset of $A$, then $A$ must be a closed set.
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Infinite intersection of frontiers

Let $(X,d)$ be a compact metric space and $f\colon X\to X$ a homeomorphism. Let $\delta >0$, define closed sets $B_n=D[f^n(x),\delta]$ (closed ball of center $f^{n}(x)$ with radius $\delta$ in ...
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31 views

Closed set, open set or neither?

Just a quick question - is a straight line that goes on indefinitely viewed as a closed set, open set or neither? Seeing as it includes all the boundary points as it travels, but it doesn't have any ...
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1answer
20 views

Quotient metric space

Let $X$ be some set, $(Y, \rho)$ be a metric space and $f:X\to Y$ be some map. Let $d$ be a pseudometric on $X$ defined by $d(x', x'') = \rho(f(x'), f(x''))$ and consider a quotient metric space ...
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Completeness of “weighted” shortest path metric

I am trying to see when this type of metric is complete: Let $A$ be the set of $C^{1}$ paths in $U \in \mathbb{R}^{n}$. For any $x,y$ define $$\rho(x,y) = \inf_{\gamma \in A; \gamma(0) = x, \gamma(1) ...
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1answer
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Proof for sets and functions.

I have been proving problems like this all day with ease, but this is is just puzzling to me. Where do I start? Also, a site with questions and answers to problems like these.
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1answer
58 views

Is an open ball a complete metric space?

Is an open ball $K((0,0),1) \subset \Bbb R^2$ with maximum metric a complete metric space? While I believe I understand basic metric space concepts I just don't have an idea how to prove or ...
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24 views

Is there a metric on the extended reals which yields regular and infinite limits?

The question is in the title: Is there a (extended) metric on the extended reals which yields regular and infinite limits? but in particular I want know the explicit construction of said ...
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Let $A$ be a subset of $\Bbb{R}$ such that the following $7$ sets are all different [closed]

I am suppose to come up with an example of an subset $A$ such that the sets $ A$ $int(A)$ $cl(A)$ $ cl(int(A))$ $ int(cl(A))$ $int(cl(int(A)))$ $ cl(int(cl(A)))$ are all different. I am ...
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Prove that cl(int(cl(int(A)))) = cl(int(A))

I am suppose to show that $ cl(int(cl(int(A)))) = cl(int(A)) $ and also that $int(cl(int(cl(A)))) = int(cl(A)) $ and I am having problems doing that becuse i just cannot figure out were to ...
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What is a contraction on a space $(X,d)$?

I have been reading some proofs on the elementary theorems of differential equations. One such proof uses the concept of a "contraction". See the definition below. Definition 4 Let $(X,d)$ be a ...
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show that $x \in A^o$ if and only if $d(A^c,x) > 0$

show that $x \in A^o \iff d(A^c,x) > 0$ where $d(A^c,x) = \inf_{y\in A^c} \lambda (x,y)$ where $\lambda$ is a metric and $(X,\lambda)$ is a metric space and $A^o$ is the set of interior points of A ...
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1answer
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Connection between weak topology in probability and weak* topology in functional analysis

In functional analysis, Definition A: for any normed linear space $(X, \| \cdot \| )$, the weak star topology $\sigma (X^*, X)$ on $X^*$ is generated by the collection of seminorms $\{ p_x ...
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32 views

written set of functions as a union of Borel measurable set

Denote by $\mathcal{H}$ the set of bounded and continuous maps $ h:\mathbb{R}\rightarrow \mathbb{R}$ such that $h(0)=0$. I wonder if you can write $\mathcal{H}$ as (not trivial) $F_{\sigma}$ set in ...
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34 views

example showing Minkowski distance with $p<1$ is not a metric

The Minkowski distance: $$\left(\sum_i |x_i-x_i'|^p \right)^{1/p},\ \text{where}\ p\ge1$$ is only a metric for $p\ge1$. Can someone give me a quick example why the triangle inequality doesn't hold in ...
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2answers
68 views

Finding all metrics of set $X=\{1,2,3\}$

I have the following problem where I'm lost a bit. Let $X=\{1,2,3\}$ and $(X,d)$ be a metric space. List all the metrics $d$ of $X$ and show that they are equivalent. (Hint: construct a ...
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1answer
55 views

Proving that if a function is a metric then it is symmetric and non negative

I am trying to prove that given a metric d using only the properties that it $d(a,b)=0 iff a=b$ and $d(a,c)\le d(a,b)+d(b,c)$ that $d(a,b)=d(b,a)$ and $d(a,b) \gt 0$ I understand that it is part of ...
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53 views

Which metric to use to make the sequence 1, 1.4, 1.414, 1.4142, .. converges in space Q?

In space Q, with the metric it inherits from R, the sequence 1, 1.4, 1.414, 1.4142, ... does not converge. Is there a way to change the metric to make it converge in Q?
2
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1answer
49 views

show that $d(X,x) = d([X],x)$

show that $d(X,x) = d([X],x)$ where $d(X,x) = \inf_{y \in X} \lambda (x,y)$ where $\lambda$ is a metric and $[X] = \{ x \in X : d(X,x) = 0 \}$ I have shown $d([X],x) \leq d(X,x)$ I am stuck proving ...