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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Doubt related to the extended real line and distance/metric

I am studying real analysis and I am being introduced to functions that take values on the extended real line, I have a fundamental doubt about this so I'll give an example to illustrate my confusion: ...
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If $g: Y \to Z$ is a continuous injection, then a map $f : X \to Y$ is open if $g\circ f$ is open.

If $g: Y \to Z$ is a continuous injection, then a map $f : X \to Y$ is open if $g\circ f$ is open. To show the map $f : X \to Y$ is open, we first take any open subset $U$ from $X$ and then show that ...
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The square $S := [- R, R] \times [-R, R]$ is a compact subset of $\Bbb R^2$.

The square $S := [- R, R] \times [-R, R]$ is a compact subset of $\Bbb R^2$. An intuitive approach: Let $S$ be not compact then there is an open cover of which there is no finite sub cover of $S$.Now ...
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28 views

Sequence characterization of bounded sets

If $M$ is an arbitrary metric space, the following holds: $A\subseteq M$ is totally bounded $\Leftrightarrow$ Each sequence in $A$ contains a Cauchy subsequence. Additionally, for ...
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25 views

Is $\overline{\mathbb{R}}^+$ a compact Polish space

if $X$ is defined by $$X= [0,+\infty)\cup\{+\infty\}$$ is endowed with the metric $$d_X(x,y) = |\arctan(x) - \arctan(y)|$$ Is it true that the metric space $(X,d_X)$ meets the following properties? ...
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26 views

Alternate definition limit

The definition of the limit of a sequence is: $L=\lim\limits_{n\to\infty}f(n)\Leftrightarrow\forall\epsilon>0:\exists N\in\mathbb{N}:\forall n\in\mathbb{N}:\left(n>N\Rightarrow ...
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Name of the metric: $d(f,g)=\max \limits_{a\leq x \leq b} |f(x)-g(x)|$

What is the name of the metric: $$d(f,g)=\max \limits_{a\leq x \leq b} |f(x)-g(x)|$$ Where $f,g\in X$ where $X$ is the space of all continuous functions. I can't find any documentation on this ...
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28 views

Checking my understanding of the Interior of these intervals

Let $[a,b]$ be any finite closed interval. (i) $\text{Int}_{[a,b]}(a,b]$ Am I correct to say that the interior of this set is $[a,b]$? Since the interior of a set are all the points in the set in ...
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33 views

An example of a dense and co-dense set in a metric space with countable derived set

Let $(X,d)$ be a metric space and $A\subset{X}$ such that $A$ and $A^c$ are both dense in $X$. Show that it is not necessary that $A^\prime$ be uncountable. And prove $(A^\prime)^\prime=A^\prime$. ...
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If $A$ is connected , $B$ is open and closed and $A \cap B$ is non empty then $A \subset B$.

Let $(X,d)$ be a metric space and $A,B \subset X$. If $A$ is connected , $B$ is open and closed and $A \cap B$ is non empty then $A \subset B$. I tried it with proving a contradiction if we first ...
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22 views

Show that a point $ a \in X$ is not a cluster point of $X$ if $a$ is isolated.

A point $a$ in a metric space $X$ is said to be isolated if and only if $r> 0$ so small that $B_r(a)$ = {$a$} Show that a point $ a \in X$ is not a cluster point of $X$ if $a$ is isolated. proof: ...
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34 views

A complete subset of a metric space is closed?

Supposing $A$ is a subset of a metric space $S$, it is simple enough to show that if $S$ is complete and $A$ is closed, that $A$ is complete. However, without being given that $S$ is complete, what ...
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51 views

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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19 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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17 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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53 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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34 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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28 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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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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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 $\{ x \in \mathbb R^n : ...
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$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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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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26 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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32 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} ...
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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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79 views

Prove that $d((x|y),(u|v)) = max \{d_{x}(x|u),d_{y}(y|v)\}$

$d_{x}$ is a metric on the set $X$. $d_{y}$ is a metric on the set $Y$. Prove that $$d((x|y);(u|v)) = max \{d_{x}(x|u),d_{y}(y|v)\}$$ defines a metric on the set $X \times Y$. I did the following: ...
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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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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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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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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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29 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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$f: \mathbb R \to \mathbb R$ 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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27 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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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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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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31 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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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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Concept of Boundedness

I noticed there are two notions of boundedness, one in the context of order theory and other in the context of metric spaces. In a metric space (X,d) , we talk about subsets of X being bounded iff ...
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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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41 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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32 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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33 views

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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1answer
15 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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75 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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1answer
22 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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1answer
28 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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1answer
39 views

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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1answer
45 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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1answer
36 views

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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1answer
17 views

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 ...