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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Isometry of a metric space with proper subset

In Irving Kaplansky's "Set Theory and Metric Spaces", exercise 17 on page 71 asks for an example of a metric space which is isometric to a proper subset of itself. Any infinite discrete space and any ...
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244 views

What is the best way to define the diameter of the empty subset of a metric space?

This question is related to Why are metric spaces non-empty? . I think that a metric space should allowed to be empty, and many authorities, including Rudin, agree with me. That way, any subset of a ...
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Is $([0, \sqrt 2] \cap \mathbb Q) \subset \mathbb Q$ closed, bounded, compact?

As far as I can tell it is bounded, as it's within $[0, \sqrt 2]$, and is closed as there cannot be an open neighbourhood about 0, and as it's closed and bounded it is therefore compact. However I'm ...
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Prove that $d_{1}$ and $d_{2}$ induce the same topology

Suppose $d_{1}(x,y) = |x-y|$, $d_{2}(x, y) = |\phi(x) - \phi(y)|$, where $\phi(x) = \frac{x}{1 + |x|}$. Prove that $d_{1}$ and $d_{2}$ are metrics on $\mathbb{R}$ which induce the same topology. ...
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Closed or open set? And Cantor metric

Since this semester I started topology. I find this extremely difficult, it takes me much time to understand anything from it. Please correct me if I am wrong. On $R^2$ space every continuous ...
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75 views

interior of a nested increasing union over a sequence of sets

What are the weakest hypotheses on a topological space $X$ so that for every increasing sequence $S_n$ of subsets of $X$ we have that $\cup _{i=1}^\infty (S_n^o)=(\cup _{i=1}^\infty S_n)^o$
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Are all isometric constant displacement maps bijective?

Let $(M,d)$ be a metric space. An isometry is a distance preserving map. A constant displacement map is a function $f$ such that $d(x,f(x)=d(y,f(x))$ for all $x$ and $y$. I know that not all ...
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Equal balls in metric space

Let $x$ and $y$ be points in a metric space and let $B(x,r)$ and $B(y,s)$ be usual open balls. Suppose $B(x,r)=B(y,s)$. Must $x=y$? Must $s=r$? What I got so far is that: $$r \neq s \implies x \neq ...
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Completing a metric space

This is captured from a chapter talking about completion of metric space in Real Analysis, Carothers, 1ed: Definition of completion: Actually, I cannot understand some parts of the proof. ...
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uniformly convergent sequence of functions on a compact space

There's an exercise on Kaplansky's textbook that says: Let $\{f_i\}$ and $f$ be continuous real functions on a compact metric space $M$. Prove that $f_i$ converges uniformly to $f$ if and only if the ...
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if a map and its inverse are continuous, does that imply injection?

I've proved that a mapping of one topology to another and its inverse are both continuous. so since f and f inverse are continuous, can I therefore say that they're injective?
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Prove $\{(x,y) \in \mathbb R^2 | 0 < x^2 + y^2 < 1 \}$ and $\{(x,y) \in \mathbb R^2 | x^2 + y^2 > 1 \}$ are homeomorphic to each other

I have $\{(x,y) \in \mathbb R^2 | 0 < x^2 + y^2 < 1 \}$ and $\{(x,y) \in \mathbb R^2 | x^2 + y^2 > 1 \}$ and need to prove they are homeomorphic to each other. I wanted to use the function ...
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Prove that $\{(x,y) \in \mathbb R^2 | y = x^2 \}$ is not compact

I know I need to choose an open cover and then show it has no finite subcover. If I use $((-n,-n^2),(n,n^2)) \forall n \in \mathbb N$ does this work?
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Are all continuous bijective translations isometries?

Let $(M, d)$ be a metric space. I define a translation on $M$ to be a function $f$ from $M$ to $M$ such that $d(x,f(x))=d(y,f(y))$ for all $x$ and $y$ in $M$. In a previous question, I asked if every ...
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38 views

Are all metric translations isometries

Let $(M, d)$ be a metric space. I define a translation of $M$ to be a function $f$ from $M$ to $M$ such that $d(x, f(x)) = d(y, f(y))$ for all $x$ and $y$ in $M$. My conjecture is that every ...
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77 views

Infinite dimensional euclidian space with the product topology metrizable?

Let $\mathbb{R}^{\omega}$ be the space of real sequenes with the product topology. Is $\mathbb{R}^{\omega}$ metrizable?
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Limit of sums is sum of limits in a metric space

So I'm aware that in a normed space, the limit of the sums is the sum of the limits: For normed space $(X, ||.||)$, if $x_n \rightarrow a$ and $y_n \rightarrow b$, then $(x_n + y_n) \rightarrow ...
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Proving that $A_f(x) = \lambda \int_a^x f(t) dt$ is a contraction

Let $C[a,b]$ be the set of all continuous functions $f:[a,b] \rightarrow [a,b]$ where $a,b \in \mathbb{R}$ and $d(f,g)= \max_{x \in [a,b]} \vert f(x)-g(x) \vert $ $\forall$ $f,g \in C[a,b]$. I had to ...
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172 views

Proving that a function is a contraction

The question is: Find values of $a$ such that the function $f(x)=ax^2 -1$ is a contraction on the interval $[1,2]$. I looked up the definition of a function being a contraction on the interval and ...
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161 views

Closure of equicontinuous family of bounded functions.

Let $B(x,y)$ be the set of all the bounded functions $f: X \to Y$ ($X,Y$ metric spaces). Prove that if $\mathcal F \subset B(x,y)$ is an equicontinuous family, then $\overline {\mathcal F}$ is ...
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81 views

How to show the set F of all finite sequences is connected in the space c0?

Question: How one can show that the set F of all finite sequences (i.e after n, the entries are zero) is connected in the space c0 (i.e. the space of all sequences that converge to zero) the metric ...
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How to show that the space $\mathcal{l}^1$ is connected

How can one show that the space $\mathcal{l}^1$ is connected? $\mathcal{l}^1$ is the metric space of real sequences such that the sum of the absolute values of the entries converges. with metric ...
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If a set $A$ is disconnected in $(X,d_1)$, then it is disconnected in $(X,d_2)$ for any metric $d_2\geq d_1$

If a set $A$ is disconnected in metric space $(X,d_1)$, then it is disconnected in $(X,d_2)$ for any metric $d_2\geq d_1$ we need to prove or disprove. we think it is true. any open set for $d_1$ is ...
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150 views

how to show a countable space is totally disconnected for any metric?

Suppose X is countable. We need to show that for any metric d on X the space (X,d) is totally disconnected. It is true that any subset of a countable set is countable. so, divide the space until its ...
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109 views

What's the real name for these things? Categories whose morphisms have “length.”

A fairly obvious "categorification" of metric spaces is as follows. First, let us agree to view $\mathbb{R}_+$ as an ordered Abelian monoid, where by "Abelian monoid" we really mean a category whose ...
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60 views

Compactness and Convergence of Subsequences

Let $(X,\rho)$ be a metric space. Suppose that $(x_n)_{n\in\mathbb Z_+}$ is such a sequence in $X$ that any subsequence has a further subsequence that is convergent. However, the limits of these ...
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29 views

Prove that $Lip [a,b]^{\circ}=\emptyset$

Let $Lip[a,b]=\{f \in C[a,b] : \exists k>0, |f(x)-f(y)|\leq k|x-y|\}$, Prove that $Lip[a,b]^{\circ}=\emptyset$ in $C[a,b]$. Suppose there exists $f \in Lip[a,b]^{\circ}$, then $\exists ...
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Every finite metric space is discrete

I know that the question is so easy, but I don't know how to conclude. I have that: Let $X$ be a finite metric space. Suppose that $X$ isn't discrete, then exists $x\in X$ such that, $\forall ...
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Diameters of and distances between sets in metric spaces

I know that: If $\DeclareMathOperator{\diam}{diam}(X,d)$ is a metric space and $A\subset X$ is bounded, then there $\sup \{ d(a,a'):a,a'\in A \}$, called the diameter of the set $A$ and is denoted by ...
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Prove it doesn't exist any function f:R→R that is continuous only at the rational points.

Prove it doesn't exist any function $f:\mathbb R \to \mathbb R$ that is continuous only at the rational points. Suggestion: For every $n \in \mathbb N$, consider the set $U_n=\{x \in \mathbb R : ...
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Derivation of Symmetry Property of Metric Spaces

I am given the following modified triangle inequality property of metric spaces, where for any $x_1$, $x_2$, $x_3 \in X$, we have $d(x_1, x_2) \le d(x_1, x_3)+d(x_2, x_3)$. I am tasked to show that ...
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Find Weight for minimum Manhattan Distance

Let's say, I have three points $(1, 4)$, $(4, 3)$ and $(5, 2)$. I need to find weight $w_1$ and $w_2$ so that the point $(1, 4)$ be the centroid of the points in ...
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Is such an infinite dimensional metric space, weakly contractible?

We counteract this answer by adding the rigidity assumption: Is there still a counterexample? Let $(X_{n},d_{n})_{n \in \mathbb{N}}$ be a sequence of complete geodesic metric spaces such that: ...
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Counterexample to: if $1\le p<q<\infty$, then $L^q(X)\subset L^p(X)$ with $\mu(X)=\infty$

We know if $\mu(X)<\infty$, and if $1\le p<q<\infty$, then $L^q(X)\subset L^p(X)$ (can be proved by using Holder's inequality). Is this still true if $\mu(X)=\infty$? Counterexample? ...
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112 views

Behavior of Hausdorff dimension under homeomorphisms

Let $X$ and $Y$ be metric spaces, $f : X\rightarrow Y$ a homeomorphism. Denote by $\dim_{\mathcal H}$ the Hausdorff dimension. I know that it is possible that $\dim_{\mathcal H} Y < \dim_{\mathcal ...
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Closed and bounded subset of a metric space which is not complete

I am trying to find a counterexample for a metric space which is not complete and has a closed and bounded subset. Any hint will be helpful. Thank you
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How does one “separate” the cartesian product properly?

Say, $\delta>0$, $X$ and $Y$ are metric spaces, $(x_0,y_0)\in X \times Y $, and there is some property $P$ such that $$\forall (x,y) \in X \times Y: \ \ \ \ d \Big( (x_0,y_0), (x,y) \Big) < ...
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Examples of rare, meager and nonmeager sets in $\mathbb{R}$

Kreyszig Functional Analysis book presents the following definition. I'm trying to get some examples. (a) The cantor set $K$ is rare in $\mathbb{R}$ because it's closed and has empty interior so ...
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Tell that sequence $(x_n)$ converges if and only if there $n_0\in \Bbb N$ such that $x_n=x_{n_0}$ for all $n\geq n_0.$*

I do not know how to solve the following example so if any of you can help me solve. Please. The example is as follows: Let $(X,d)$ a discrete metric space and $(x_n)$ is a sequence in $X$. Tell that ...
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When is a Lipschitz homeomorphism of metric spaces bi-Lipschitz?

Let $(X,d_X)$ and $(Y, d_Y)$ be metric spaces, and let $f: X \to Y$ be a Lipschitz map which is a homeomorphism of the underlying topological spaces. Are there conditions which assure that $f$ is ...
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The characterization of asymptotic dimension

Let X be a metric space. The following conditions are equivalent (a)asdimX = n (b)n is the smallest integer such that for every R > 0 there exists n + 1 families Ui i=0,1,2,...,n, and S > 0 such ...
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Comparing different topologies

I have to solve the following: For every function $f\in C[0,1]$, $\varepsilon>0$ and every finite set $A$, set $U(f,A,\varepsilon)$ is defined by $U(f,A,\varepsilon)=\{g\in C[0,1]:\forall x\in A\; ...
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98 views

strong equivalent metrics

Let $(X,d_x),(Y,d_Y)$ be bounded metric space. Let $f:X\rightarrow Y$ be a homeomorphism. Is it true that there exist $a,b>0$ such that $$ad_X(x_1,x_2)<d_Y(f(x_1),f(x_2))<bd_X(x_1,x_2)$$ for ...
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Distance Metric in 4 dimensions $\Bbb R^3\times SO(2)$

The euclidean distance metric, $\sqrt{dx^2+dy^2+dz^2}$, shows the shortest distance between two points in $\Bbb R^3$. What would be the distance metric to show the shortest distance between two ...
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find nested closed balls of polynomials s.t. intersection is empty (in a metric space)

the space is the set of all polynomials on $[0,1]$ with metric sup. the space is not complete. we need to find explicitly a nested family of closed balls with radius (of each closed ball) goes to ...
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86 views

Different “$\pi$s” [duplicate]

Does any one know of a concept analogous to $\pi$ in metric spaces. Namely, taking the all the points $1$ away from a point, and measuring the distance as some sort of limit? This was prompted when I ...
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Diameter of a subset of a metric space

Let $(\Bbb R,d)$ be the metric space with the metric function $$d(x,y)=\frac{|x-y|}{1 + |x-y|}\;. $$ Calculate $\operatorname{diam}(0,\infty)$. I am thinking the answer is $1$ because ...
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Two equivalent definitions of convergent sequences?

I know that: Definiton 1. The sequence $(x_n)$ in the metric space $(X,d)$ is said to converge to the point $x_0\in X$ if $$\forall\epsilon>0, \exists n_0\in\mathbb{N} \text{ such that } \forall ...
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Getting a root of a continuously differentiable function by Banach's Fixed Point Theorem.

Banach Fixed Point Theorem: Consider a metric space $X = (X, d)$, where $X\neq \varnothing$. Suppose that $X$ is complete and let $T: X \to X$ be a contraction on $X$. Then $T$ has precisely one ...
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How to prove that $(\mathbb{Z}, d)$ with $d(m,n)=\vert m -n \vert$ is complete

I know that the metric space $(X,d)$ is called complete if each Cauchy sequence is convergent, but I don't now how to prove the following: $(\mathbb{Z}, d)$ with $d(m,n)=\vert m -n \vert$ is ...