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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How to show $d(Tx,Ty) \geq d(x,y)-d(x,Tx)-d(y,Ty)$?

I'm reading a proof that has this inequality which I figure follows somehow from the triangle ineq. but I can't figure it out. I can provide more details if needed but I don't want to write out the ...
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Is this sequence necessarily Cauchy?

Let $(X,d)$ be a metric space and $(x_n)$ a sequence in it such that for all $i,j,n\in\mathbb{N}_{>0}:$ $$d(x_i,x_j)<n+1\Rightarrow d(x_{i+1},x_{j+1})<n$$ ...
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Open cover of a metric space $M$ when $f(x) \neq a \in \mathbb{R}, \forall x \in M$

$f: M \to \mathbb{R}$ is a continuous function, where $f(x) \neq a \in \mathbb{R}, \forall x \in M$. I need to prove that the collection of open sets $U$ for which either $f(x) > a$ for $x \in U$ ...
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Example of two closed disjoint set $X, Y$ so that $d(X, Y) = 0$

I am looking for an example of a metric space $M$ and non empty disjoint closed subsets $X$ and $Y$ such I that $d(X,Y)=0$, where $$d(X,Y)=\inf_{x\in X, y\in Y} d(x, y).$$ I’m thinking it might have ...
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Notation for continuous functions

Let's say $(X,\sigma)$ and $(X,\tau)$ are topological spaces, and $f$ is a continuous function from the former to the latter. (That is, the inverse images of elements of $\tau$ are elements of ...
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Let $A$, $B$ be compact subsets of $\Bbb R$ and define $C = A + B =\{\, a+b\mid a \in A , b \in B\,\}$.

Let $A$, $B$ be compact subsets of $\Bbb{R}$ and define $C = A + B =\{\, a+b\mid a \in A , b \in B\,\}$. Prove that $C$ is compact. Use induction to show if $A_1, A_2,\ldots A_n$ are compact subsets ...
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Show $g=3$ is contained in the unit ball of $X=C[0,1]$ in the uniform metric.

How do I show that the function $g=3$ is contained in the unit ball of $X=C[0,1]$ in the uniform metric? I know the uniform metric $$d_u(g,0)=sup |3|$$ where do I go after this?
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Finding the uniform and $L^{1}$ metric between two functions.

I know the uniform metric between two functions f and g is defined as $$d_u (f,g)=sup|f(x)-g(x)|$$ and the $L^{1}$ is defined as $$d_1 (f,g)=\sum^{n}_{i-1} |f-g|$$ Say that $$X=C[0,1]$$ Lets say I ...
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cauchy sequence on $\mathbb{R}$

i want to show that $\mathbb{R}$ with the following metric : $d_1(x,y)=|x^3-y^3|$ is complete. I think a good way to show it is to show that a sequence which is Cauchy for $d_1$ will also be Cauchy ...
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Prove that $ U=\{f\in C[0,1]: f(x)\neq 0, \forall x \in [0,1]\}$ is open and find his connected components

In $(C[0,1],d_\infty)$, consider $U=\{f\in C[0,1]: f(x)\neq 0, \forall x \in [0,1]\}$. Prove that $U$ is open and find his connected components. I know that for proof the first thing, i have to show ...
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showing $Ax=b$ has a unique solution by finding the fixed points of a function

Let $n ∈ \mathbb N$. Consider an $(n×n)$-matrix A with real components and a column vector $b ∈ \mathbb R^n$. They give rise to an affine transformation $T : \mathbb R^n → \mathbb R^n$ with $T(x) = ...
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Is every manifold a metric space?

I'm trying to learn some topology as a hobby, and my understanding is that all manifolds are examples of topological spaces. Similarly, all metric spaces are also examples of topological spaces. I ...
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Left invariant metrics on a Lie group coming from Lie algebras

Let $G$ be a Lie group and $\mathfrak{g}$ be its Lie algebra. If we have an inner product on $\mathfrak{g}$ then we can create a left invariant metric $d_G$ on $G$ by translations. On the other hand ...
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Distinctions of different topologies on the sequence space (countable cartesian products of $\mathbb{R}$)

$\newcommand{\b}[1]{\mathbf{#1}}$ $\newcommand{\R}{\mathbb{R}}$ $\newcommand{\N}{\mathbb{N}}$ Question I solved this exercise in Munkres.(20.4) But I don't know if I did it righ t or not. I really ...
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Is the minimum of two metrics is again a metric?

Let $d_1$ and $d_2$ be two metrics on non empty set $X$. Is $d$ = $\min\{d_1, d_2\}$ is again metric on $X$? I'm looking for a counter example with minimum of two metrics not being a metric.
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All the spaces which are Hausdorff and semi metric spaces, are necesarily metric spaces?

My hypothesis: $(X,d)$ is a semi metric space and a Hausdorff space My thesis: $d$ is metric My try: Well $d$ is semi metric so, to prove that $d$ is metric, I only have to prove that: $d(x,y)=0 ...
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Cardinality of a metric space

Let $X$ be an infinite set .For any two metrics $d_1,d_2 $ on $X$ the identity map $i:(X,d_1)\to (X,d_2)$ is continuous. Prove that $X$ is always countable. I am not getting how to start this ...
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Example about Hausdorff distance

As we know Hausdorff distance for two compact sets is defined like: $$d_H (A,B)=\max\{\sup_{a\in A}(a,B), \sup_{b\in B}(b,A)\}$$ And compact set sequence $C_i$ converge to $C$ iff $ ...
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Is $A$ necessarily a compact set or a connected set?

Let $f:\mathbb R\to \mathbb R$ be a continuous function and $A\subset \mathbb R$ is defined by : $A=\{y\in \mathbb R:y=\lim f(x_n),$ for some sequence $x_n\to \infty\}$ Is $A$ necessarily a ...
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Compact convergence of inverse functions

Consider two metric spaces $X$ and $Y$ and a sequence of functions $f_n\colon X\to Y$ together with a function $f\colon X\to Y$. Assume, all $f_n$ and $f$ have inverse functions $g_n$ and $g$, say. It ...
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Function approximation without metric, is it possible?

As I learn't from analysis/functional analysis the approximation technique are based on the use of a norm in a metric space, even the interpolation is actually a particular case of these approximation ...
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How to embed points on a sphere in a 3 or 4 dimensional space

I am looking for a procedure to embed points on a sphere in a 3 or 4 dimensional Euclidean space such that the distances are preserved as much as possible. If there is any related optimization ...
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Proof for continuity

Theorem: Let $U \subseteq \mathbb{R}^m, f: U \to \mathbb{R}^n$. Then $f$ is continuous on $U \Leftrightarrow$ For all closed sets $T \subseteq \mathbb{R}^n$, there exists a closed set $S ...
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Connected componentes of $X=\underset{n\in\mathbb{N}}{\bigcup} A_n\cup \{(0,0),(0,1)\}$

I have to find the connected components of $X=\underset{n\in\mathbb{N}}{\bigcup} A_n\cup \{(0,0),(0,1)\}$, where $A_n=\{\frac{1}{n}\}\times [0,1]$, for every $n\in\mathbb{N}$. Well, first of all, i ...
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Showing $d_1(x,y)=|x-y|$ and $d_2=\sqrt{|x-y|}$ for $x,y\in\mathbb{R}$ define equivalent metrics

This isn't homework, I am just trying to better understand how to show two functions define the same metric. I would like to show that $d_1(x,y)=|x-y|$ and $d_2=\sqrt{|x-y|}$ for $x,y\in\mathbb{R}$ ...
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Link between finite measure spaces and bounded metric spaces

I was taking a look at some measure theory, and, finding the concept of finite measure space I thought about the following question: Is there some sort of a conceptual link (in the form of ...
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Unit interval in $\mathbb{Q}$ is not totally bounded.

I was reading an analysis textbook, and I came across with a theorem says that a set is compact iff it's closed and totally bounded. But if we consider the unit interval in the metric space ...
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Is $C^1([a,b])$ with sup norm a complete metric space? [duplicate]

Give $C^1([a,b])$ the sup metric induced from $(C^0([a,b]),||.||)$. I want to know if $C^1([a,b])$ is complete. My thinking is to show that $C^1([a,b])$ is a closed subset of $C^0([a,b])$. Let ...
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Showing two metrics are equivalent

I'm trying to solve the following problem. Let $d$ and $\rho$ be metrics on the same space $M$, and suppose that there exists a continuous function $\phi : [0, \infty) \to [0, \infty)$ such that: ...
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Find interior of A

I have the following exercise: Let $(X,d)$ be the metric space $\mathbb{R}^2$ with $d((x_1,y_1),(x_2,y_2))=\sqrt{(x_1-x_2)^2+(y_1-y_2)^2}$. Find the interior of $A$ and $B$ where $A=\{(x,y), x=2\}$ ...
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Definition: honest distance function

What does honest distance function mean? In the context of metric spaces.
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Locally Compact Metric Space Properties

(i) Let $X$ be locally compact, and $K$ compact. Show that, given $\delta > 0$, there exists finitely many compact closed balls of radius at most $\delta$ that cover $K$. Proof: Let $\delta ...
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Proving a map/function based on two metric spaces is continuous.

Let $X \neq \emptyset$ be a set and $d$ the discrete metric on $X$. Let $(\Xi, \delta)$ be another metric space and $f : X → Ξ$ a map. How can I show that $f$ is continuous? Thank you in advance.
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e/3 argument? Uniform convergence from a topoloical space to a metric space

I am trying to prove that if we take a sequence (say fn) in C(X,Y) that converges uniformly to to a function f: X to Y, then f must be an element of the space C(X,Y). Where f moves from a topology to ...
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Should the distance between $\{x: f(x)=0\}$ and $\{x:f(x)<0\}$ be $0$ ($f$ may not be continuous, $x$ in some metric space)?

Suppose I have a function (which may or may not be continuous) $$f: X\rightarrow\mathbb{R}$$ where $X$ is a metric space, with distance $d: X\times X\rightarrow \mathbb{R}$. By abuse of notation, we ...
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Mean and variance on a metric space

It is my first post, so please correct me if I am not following the rules/etiquette. Assume that we are given a space $\mathcal{S}$ composed by vectors $x\in\mathbb{R}^L$, constrained by $$\sum ...
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How to write a program that, given a metric, finds the locally flat coordinates at an arbitrary point?

In A. Zee's Einstein Gravity in a Nutshell on page 90, he encourages the reader to do this: Is that even possible without having a formula parsing and manipulation engine? For example the ...
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Definition of space $L_f^2$ where $f$ is a function?

http://it.tinypic.com/r/2iqjvbl/9 Hi guys! I'm writing my thesis for my degree and it's about Sturm-Liouville theory applications. I'm using the book "Al-Gwaiz M.A. Sturm-Liouville Theory and Its ...
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Basic real analysis - Relative openness theorem

I am trying to get the intuition behind the following basic real analysis theorem, with $X$ being some metric space. This is thm 2.30 from Rudin. A set $E \subseteq Y \subseteq X$ is open relative to ...
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Are there different ways to embed surface with nonvanishing curvature in a higher-dimensional Euclidean space?

The two-dimensional Euclidean space $E^2$ can be embedded isometrically in many ways in $E^3$. Its First Fundamental form will always be $\delta_{\mu\nu}$, but the Second Fundamental form $II$ will ...
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Finding closure/interior of subset of function space

Consider the subset $$A=\left\{f\in C(\Bbb R): |f(x)|< \frac{1}{1+|x|} \, \text{for all } x\in \Bbb R\right\}\subset \left\{f\in C(\Bbb R): \lim_{|x|\to \infty}f(x)=0 \right\}=X.$$ where $X$ is ...
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Contraposition of continuity

Function f is continuous on a metric space (M,d) if $$\forall x\ \forall \epsilon >0\ \exists \delta >0\ \forall y\ \ d(x,y)< \delta \Rightarrow d(f(x),f(y))<\epsilon $$ I want to find a ...
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Equivalent of metric spaces

Define $( \mathbb R^n, m) $ as a metric space such that $ m=\max \{ | x_i - y_i |: i=1...n \} .$ And also $(\mathbb R^2, d) $ as another metric space such that $d= \sum_{i=0}^n | x_i - y_i | .$ ...
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Realizing 4 point space to a euclidean space

http://www.cs.toronto.edu/~avner/teaching/S6-2414/LN1.pdf I was reading the link above and am stuck on Example 2.2. Why should f(d), f(a) and f(b) be collinear in $R^k$ ? I understand that the ...
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If $i$ is an isometry, how to prove that the image under $i$ of closed set is closed?

Suppose that $(X, d_X)$ and $(Y, d_Y )$ are complete metric spaces. Let $i \colon X → Y$ be an isometry. Suppose that $F ⊂ X$ is closed. How can I prove that $i(F)$ is a closed subset of $Y$?
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If f is continuous does this means that the image under f of any open set is open?

If $(X,d_X)$ and $(Y,d_Y)$ are metric spaces, and $f : X → Y$ is a continuous map, is it true that for any open set $U ⊂ X$, the set $f(U)$ is open in Y ?
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Do I Understand Closed Versus Complete in Metric, Normed and Inner Product Spaces?

I've looked at a number of references on this including some questions on stack exchange. Am I correct if I summarize by stating the following ? (1) A space C (metric, normed, or inner product) is ...
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1answer
51 views

Properties of a metric

Suppose that we have the space $(C(0,1), \rho)$. Then we can define a metric but this won't come from a norm. Could you explain me the above proposition? Also I want to show that if $\rho(x,y)$ is ...
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Hausdorff dimension of homeomorphic compact metric spaces

Are there examples of homeomorphic compact metric spaces of different Hausdorff dimension? If yes, are there sufficient conditions on the spaces which would imply the equality of Hausdorff ...
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Distance between two 2d similarity transformations

For 3d rotations the common way define distance between two rotations $R_0$ and $R_1$ as $d(R_0, R_1)$ = angle of $R_0 R_1^{-1}$. What would be analogue for 2d similarity transforms (rotation + scale ...