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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Does there exist non-compact metric space $X$ such that , any continuous function from $X$ to any Hausdorff space is a closed map ?

I know that there is a topological space $X$ which is not compact but such that , for any Hausdorff topological space $Y$ , any continuous function $f:X \to Y$ carries closed sets to closed sets . I ...
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76 views

Let $X$ be the union of axes is it homeomorphic to a line, a circle, a parabola or the rectangular hyperbola $xy = 1$?

Let $X$ be the union of axes given by $xy = 0$ in $\Bbb R^2$ . Is it homeomorphic to a line, a circle, a parabola or the rectangular hyperbola $xy = 1$? If we remove the origin from the union of axes ...
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171 views

Evenly Spaced Integer Topology is Metrizable

Fustenborg's proof uses an evenly spaced integer topology on $\mathbb Z$ which declares that a basis of open sets as those of the form $a + b \mathbb Z$ (i.e. arithmetic progressions). I'm interested ...
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131 views

Is there a metric proof for the Caristi theorem?

I'm writing a paper about hyperconvexity in metric spaces and came across Caristi's theorem: Let $(X, d)$ be a complete metric space and $\phi\colon X \to \mathbb{R}^+$ a continuous function. An ...
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175 views

Show that a Cauchy sequence has a fast-Cauchy subsequence

A sequence $\{x_j\}$ is said to be fast-Cauchy if $\sum_1^\infty d(x_j,x_{j+1})<+\infty$. Show that every Cauchy sequence has a fast-Cauchy subsequence. **My attempt:**Argue by contradiction, ...
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73 views

How many degrees of freedom are in a flat metric and how does one count them?

I think that there are zero degrees of freedom in a flat metric, but I do not know how to count them. I know that any symmetric metric tensor has $n(n-1)/2$ degrees of freedom, where $n$ is the ...
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61 views

How to find an open ball for a metric space?

I don't understand the process to find the open ball. I understand the definition and I understand that for B(0, delta), I need to substitute x as 0. After this stage, I don't understand where to go ...
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44 views

Help needed in a proof to show path connected-ness of a special kind of set in real Euclidean space

Let $A$ be a connected subset of $\mathbb R^n$ and $\epsilon >0 $ , consider $V:=V(\epsilon , A):=\{x \in \mathbb R^n : d_A(x)< \epsilon \}$ , I need to show that $V$ is path connected , so we ...
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90 views

Is every open cover of $\{(x,y) \in \mathbb R^2 : x^2+y^2=1\}$ also an open cover of some $\delta$ annulus around the unit circle?

Let $\{U_\alpha\}$ be an open cover of $\{x \in \mathbb R^n:\|x\|=1 \}$ , $n \ge 2$ , then does there exist $\delta >0$ such that $\{U_\alpha\}$ is also an open cover of $\{ x \in \mathbb R^n : 1-\...
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50 views

$f : U \to \Bbb R$ be a differentiable function such that $D f (p) = 0$ for all $p \in U$. Then $f$ is a constant function.

Let $U$ be an open connected subset of $\Bbb R^n$ and $f : U \to \Bbb R$ be a differentiable function such that $D f (p) = 0$ for all $p \in U$. Then $f$ is a constant function. I am facing ...
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42 views

Given any two points $x, y \in X$ there exists connected set $A$ such that $x, y \in A$. Then $X$ is connected.

Let $X$ be a (metric) space such that given any two points $x, y \in X$ there exists connected set $A$ such that $x, y \in A$. Then $X$ is connected. Let us consider a continuous function $f : X \to \...
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54 views

Completion of a sequence space

Let $F$ be a field with some absolute value $|\cdot|$. Consider the space $X$ of sequences $\mathbf{a} = (a_1, a_2, a_3, \cdots)$ for which $a_i \in F$ for all $i\in\mathbb{N}$ and at most finitely ...
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104 views

Distance on riemann sphere [duplicate]

Let we have $C$ the set of complex numbers and $z_1 , z_2 \in C $ we have $Z_1 , Z_2 \in S$ correspond on riemann sphere and we will define : $$ d(Z_1,Z_2)=\frac{2|z_1-z_2|}{\sqrt{1+|z_1|^2} \sqrt{1+|...
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Continuity over a compact subset of a metric space implies continuity everywhere

Let $f: (X, d_X) \rightarrow (Y, d_Y)$ be a function from metric spaces. If $f$ restricted to any compact subset of $X$ is continuous, then $f$ must be continuous everywhere. Should I proceed with ...
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63 views

Let $A$ be a connected subset in $ \Bbb R^n$ and $ \epsilon >0$ then the $ \epsilon$ -neighbourhood of $A$ is path-connected.

Let $A$ be a connected subset in $ \Bbb R^n$ and $ \epsilon >0$ then the $ \epsilon$ -neighbourhood of $A$ defined by $U_\epsilon(A) := \{x \in \Bbb R^n : d_A(x) < e\}$ is path-connected. If $...
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1answer
111 views

Is $\{(x,y) \in \mathbb R^2 : xy=0 \}$ homeomorphic to $\mathbb R$?

Is $\{(x,0) : x \in \mathbb R \} \cup \{(0,y) : y \in \mathbb R \}$ homeomorphic to $\mathbb R$ ? I am totally stuck and I don't even have any intuition whether they should be homeomorphic or not . ...
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61 views

Let $X$ be the union of open disk in $\Bbb R^2$ along with the tangent line $x =1$ then X is connected.

Let $X$ be the union of open disk in $\Bbb R^2$ along with the tangent line $x =1$ then X is connected. Here I use the following criterion for $X$ to be connected: A metric space $(X,d)$ is ...
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45 views

Help creating a more insightful proof looking at closures of a metric space

My lecture notes from my metric space course contained the following practice questions. I am getting very confused by this question because I found the following statement on wikipedia "A metric ...
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30 views

Question about convergence in a metric space

For part a) my strategy was showing that since E is sequentially compact, by the Borel-Lebesgue theorem it is compact. For part b) I am not sure how to solve the problem. Can I simply use the ...
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23 views

$f: \mathbb R \to \mathbb R$ as $f(x)=\sin (1/x) , \forall x >0 ; f(x)=0 , \forall x \le 0$ , is the graph of $f$ connected in $\mathbb R^2$?

Consider the function $f: \mathbb R \to \mathbb R$ as $f(x)=\sin (1/x) , \forall x >0 ; f(x)=0 , \forall x \le 0$ , then $f$ is not continuous on $\mathbb R$ . Is the graph of $f$ i.e. $G(f) :=\{ (...
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40 views

Equivalent distances

I am interested in the following property about distances: Given two distances $d_1$ and $d_2$, $$ d_1(x,y_1) < d_1(x,y_2) \Leftrightarrow d_2(x,y_1) < d_2(x,y_2). $$ Under my point of view, ...
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24 views

Embedding of $K_{2,3}$ into $\ell_1$

I am looking for hints for the following problem: Prove that every embedding of $K_{2,3}$ (with the shortest path metric and unit edge-length) into $\ell_1$ has distortion at least 4/3! Notation: $\...
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63 views

A measure on the space of probability measures

I've been reading about optimal transport and it's connections to geometry. At some point one has to study a bit of the structure of the space of probability measures, $\mathcal{P}(X)$, (over a metric ...
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142 views

Show that $R$ is closed but not sequentially compact.

Show that $R$ is closed but not sequentially compact. Attempt: A subset E of a metric space X is said to be sequentially compact if and only if every sequence $x_n \in E$ has a convergent ...
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1k views

Every sequentially compact set is closed and bounded.

A subset $E$ of $X$ is said to be sequentially compact if and only if every sequences $x_n \in E$ has a convergent subsequence whose limit belongs to $E$. Prove that every sequentially set is closed ...
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If $\operatorname{id}:(X,d_1) \to (X,d_2)$ is continuous for any two metrics $d_1$ and $d_2$, then what will be $X$?

Let $X$ be a set with the property that for any two metrics $d_1$, and $d_2$ on $X$, the identity map $\operatorname{id} : (X, d_1) \to (X, d_2)$ is continuous. Which of the following are true? (...
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43 views

Given two balls and a point show there radii $c,d$ such that $B_c(x) \subseteq B_r(a) \cap B_s(b) $

Show that given two balls $B_r(a)$ and $B_s(b)$, and a point $x \in B_r(a) \cap B_s(b)$, there are radii $c$ and $d$ such that $B_c(x) \subseteq B_r(a) \cap B_s(b) $ and $B_d(x) \supseteq B_r(a) \...
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256 views

Show that the metric space C[a,b] is complete. [duplicate]

Prove that the metric space $C[a,b]$ is complete. Where $C[a,b]$ is the collection of continuous $f:[a,b] → R$ and $||f|| = sup_{x \in [a,b]} |f(x)|$, such that $\rho (f,g) = ||f - g||$ is a metric ...
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Let $U$ be an open connected subset and $f : U \to \Bbb R$ be a diff function then $f$ is a constant function.

Let $U$ be an open connected subset of $\Bbb R^n$ and $f : U \to \Bbb R$ be a differentiable function such that $D f (p) = 0$ for all $p \in U$ then $f$ is a constant function. If we can prove that $...
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30 views

Surjectivity of isometry

I am looking for the proof Prove of "any isometry S is a surjective mapping". My attempt: pick any two points $A, B$, consider their images $S(A) = A'$ , $S(B) = B'$ . To prove surjectivity, I need ...
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100 views

If $id:(X,d_1)\to (X,d_2)$ is continuous then what will be $X$?

Let, $id:(X,d_1)\to (X,d_2)$ is continuous. Then which is(/are) TRUE ? (A) $X$ must be singleton. (B) $X$ can be any finite set. (C) $X$ can NOT be infinite (D) $X$ may be infinite but NOT ...
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58 views

Proving a homeomorphism when graph of function has product topology

Suppose $f : (X,d_x) \rightarrow (Y,d_y)$ is a function between metric spaces, and $X \times Y$ has the product topology. The graph $G_f$ is the subspace $G_f = \{(x,f(x)) \mid x \in X\}$. Define $...
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226 views

If two sequences converge, then the sequence of distances between them also converges

Question: Let $(X,d)$ be a metric space, and let $(a_{n})$ , $(b_{n})$ be convergent sequences in X with limit a, b respectively. Prove that $$(d(a_{n}),(b_{n}))$$ is a convergent sequence in $\...
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Relationsip between two definitions of the christoffel symbol?

When I first started learing about tensor calculus, the professor defined the Christoffel symbol as $$\Gamma ^a _{bc} = Z^a \cdot \partial_b Z_c $$ Where $Z^a$ is a contravariant basis vector and $...
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52 views

Equivalent Metric Using Clopen Sets

Prove that if $(X,d)$ is a metric space and $C$ and $X \setminus C$ are nonempty clopen sets, then there is an equivalent metric $\rho$ on $X$ such that $\forall a \in C, \quad \forall b \in X \...
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Prove the triangle inequality for d(x,y) = min(|x−y|,1−|x−y|)

Let X be the set [0,1). Define a non-standard metric on X as follows: For two numbers x,y ∈ X, take d(x,y) = min(|x−y|,1−|x−y|). Show that this is a metric. In order to show this is a metric, I need ...
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Counterexample for continuous function over product topology without compactness

Suppose $f$ $(X,d_x)$: $\rightarrow$ $(Y,d_y)$ is a function between metric spaces, and $X \times Y$ has the product topology. The graph $G_f$ is the subspace $G_f$ = {$(x,f(x))$ | x $\in$ $X$}. If $...
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$f :\mathbb N \to \mathbb R$ be the function $f(0)=0 , f(n)=\dfrac 1 n , \forall n >0$;is $\mathbb N$ induced with the metric $|f(x)-f(y)|$ compact?

Let $\mathbb N$ be the set of non-negative integers and $f :\mathbb N \to \mathbb R$ be the function $f(0)=0 , f(n)=\dfrac 1 n , \forall n >0$ , then obviously $f$ is injective , so $d : \mathbb N \...
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Which complete weighted graphs are obtained from finite metric spaces?

Let $(X, d)$ be a finite metric space with $X = \{x_1, \dots, x_n\}$. We can associate to this metric space a complete weighted graph with vertices labelled by the points of $X$, and edges weighted by ...
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Order and Metric

Consider the Reals as a totally ordered set via its natural order ( linear continuum ). Such order induces an order topology ( basis is the collection of open-intervals of that order), which ...
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Convergence of sequence of compact sets in Hausdorff metric

Given a sequence of compact sets $K_{i}$ in $\mathbb{R^{n}}$ and a compact set $K$ in $\mathbb{R^{n}}$, which satisfy the following 2 conditions. $\forall$ $x$ $\in$ $K$, $\exists$ $x_{i}$ $\in$ $K_{...
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191 views

Why does countable compactness imply compactness on metric spaces?

By "$E$ is countably compact", I mean that every countable open cover of $E$ has a finite subcover. By "$E$ is compact", I mean that every open cover of $E$ has a finite subcover. Let $M$ be a metric ...
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182 views

is union of nested compact spaces still compact?

Stel $D$ a metric space. Let $K_1 \subset K_2 \subset K_3 \subset ...$ a serie of compact sets in $D$. I was wondering if $K = \bigcup_{n=1}^\infty K_n$ is compact too. If we take an open cover of $K$ ...
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68 views

Total order and its order topology

I noticed that the natural order of the Reals alone, being complete ( satisfying LUB ) , is able to prove that the induced order topology is complete ( every cauchy sequence converges ). We are ...
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106 views

Proving set of bounded continuous functions is an open set

appreciate your help with the below: Question: Let C[0,1] be the set of continuous functions from [0,1] to $\mathbb{R}$. Consider the metric space M = (C[0,1],d) where d denotes the sup metric. ...
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1answer
37 views

The distance distribution from the mean for an n-dimensional normal(Gaussian) distribution

Let's say we have an n-dimensional normal distribution with identity covariance matrix and 0 mean. When we draw random points in this distribution, how do I get the distribution of the distance from ...
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64 views

If $E= A\cup B \cup C$ and $E$ is connected , where $A$ and $B$ are disconnected and $C$ is connected, then $A \cup C$ is connected.

If $E= A\cup B \cup C$ and $E$ is connected in a metric space $(X,d)$, where $A$ and $B$ are disconnected and $C$ is connected, then $A \cup C$ is connected. If we consider that $A \cup C$ is not ...
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476 views

Show that the discrete topology on $X$ is induced by the discrete metric

Let $X$ be a set. Show that the discrete topology on $X$ is induced by the metric $d(x, y) = \left\{ \begin{array}{ll} 1 & \mbox{if } x \neq y \\ 0 & \mbox{if } x = y \end{array} \...
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83 views

Increasing sequence of open sets in a separable metric space.

Suppose X is a separable metric space and ($U_α$ : α < γ) is an increasing sequence of open sets (i.e. $U_α$ ⊆ $U_β$ for α < β). Show that there is a countable $γ_0$ such that $U_α$ = $U_β$ for ...
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29 views

Weak Convergence in Metric Space proof

I have been reading Billingsleys book where I came across this theorem and proof. I am having difficulty understanding the theorem/proof. I feel there is a better, more complete way to prove it. Does ...