Metric spaces are sets on which a metric is defined. A metric is a generalisation of the concept of "distance". Metric spaces should not be confused with topological spaces.

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

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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15 views

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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29 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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0answers
42 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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264 views

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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26 views

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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78 views

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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65 views

Prove that the following statements are equivalent characterizations of continuity

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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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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25 views

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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40 views

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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60 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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62 views

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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75 views

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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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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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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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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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?
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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 ...
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Is Topological Space a Metric Space?

What's the correct relationship between these two spaces? I think that topological space is a metric space, since the open is defined by a metric such that $d(x, a) < \epsilon$.
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Why is $[-1,1]$ compact when $a_n = (-1)^k$ does not converge in $A$

I know this question sounds silly but I was reading the definition of compactness and couldn't quite wrap my head around this Compactness :A subset $A$ of a metric space $M$ is compact if every ...
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Is $T$ a homeomorphism?

Let $X$ be the space of all polynomials in one variable over $\Bbb R$. If $p=a_0+a_1x +a_2 x^2+...+a_n x^n$,define $||p||=|a_0|+|a_1|+...+|a_n|$. Which are correct? $(X,d)$ is complete where ...
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Why is the function $f(x)=x^2$ is not a contraction on $[0,0.5]$?

Let $(X,d)$ be a metric space and let $F:A(\subset X)\to X$. We say $F$ is a contraction if there exists $\lambda$ where $0\leq\lambda<1$ such that $$d(F(x),F(y))\leq\lambda d(x,y)$$ for all ...
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2answers
43 views

How can one measure distance between point and the line in maximum metric space?

Given metric space $M = (\mathbb{R}^2, d)$ where $d = \operatorname{max}\{|x_1 - y_1|, |x_2 - y_2|\}$, how can one measure distance from some arbitrary point $X$ to the line $y = 3$, let's say? How ...
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If $X \setminus A$ is disconnected then prove or disprove $X \setminus B$ is also disconnected

Let $X$ be a connected metric space ( with more than one point ) and $A \subseteq X$ be not closed in $X$ and such that $X \setminus A$ is not connected ; then is it true that $X \setminus B$ is also ...
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Distance geometry and pythagorean theory. Pairwise distances to absolute 2D coordinates

I don't have sufficient mathematical background. I am trying to get the absolute 2D coordinates from the pairwise comparison distances: What I have distances between points: p1-p2 = 0.3 p1-p3 = ...
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1answer
32 views

Proving that $d(a,b)=p^{-n}$ is a metric for $\mathbb{Q}$

I have the following task: If we have the metric $d:\mathbb{Q}\times\mathbb{Q}\rightarrow\mathbb{R}$, so that $d(a,a)=0$ and $d(a,b)=p^{-n}$ always when $a-b=p^nh/k$, where ...
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Why is the metric on $\mathbb{N}$ defined as the following?

This is from Muscat's Functional Analysis:http://staff.um.edu.mt/jmus1/metrics.pdf Show that $d(m,n) = |\dfrac{1}{m} - \dfrac{1}{n}|$, $m,n \in \mathbb{N}$ So the first two properties of the metric ...
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Let $(X,d)$ be a metric space and $Y\subset X$. Suppose $G\subset X$ is open; show that $G\cap Y$ is open in $(Y,d)$.

Let $(X,d)$ be a metric space and $Y\subset X$. Suppose $G\subset X$ is open; show that $G\cap Y$ is open in $(Y,d)$. I'm not sure how to show this result. Any solutions/hints are greatly ...
2
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1answer
80 views

Does every connected metric space $X$ contains a connected subset $A$ such that $X \setminus A$ is infinite?

Convention : Whenever we are going to talk about connected spaces , we will mean with more than one point . I am trying to see whether every connected metric space $X$ contains a connected subset ...
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Existence of a special kind of continuous injective function $f\colon A \to \mathbb R$, where $A$ is countable, relating to connectedness

Let $A \subseteq \mathbb R$ be a countable set ($A$ induced with usual subspace topology), then does there necessarily exist a continuous injective function $f\colon A \to \mathbb R$ such that for ...