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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Existence of metric $d$ on $\mathbb R$ such that the function $f:(\mathbb R,d) \to (\mathbb R,d)$ ; $f(x)=-x$ is everywhere discontinuous

Does there exist a metric $d$ on $\mathbb R$ such that the function $f:(\mathbb R,d) \to (\mathbb R,d)$ defined as $f(x)=-x$ is everywhere discontinuous ? It is motivated from this question which ...
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A set is compact iff every collection… Proof check

I asked this question (A set is compact iff all closed collections of subsets with the f.i.p. have nonempty intersection) a few days ago and was lucky enough to get an answer, but I'm afraid that the ...
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Infinite Union of Complete Metric Subspaces which is Not Complete

Can anyone think of an example of a metric space $(X,d)$ and an infinite set of complete metric subspaces in $(X,d)$ such that their union is not complete?
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Example Infinite Product of Compact Metric Spaces that is not itself compact [closed]

Can anyone think of an infinite product space of compact metric spaces that us not itself compact?
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Showing this a norm

I want to show that $$\| x \| = \sum_{n=1}^{\infty} \frac{1}{2^n} \frac{\left| x_n \right|}{1+\left| x_n \right|}$$ is a norm. I'm fine showing positivity and the triangle inequality, to show the ...
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$\ell_0$ norm and the induced complete metric spaces

I have been reading about the $\ell_0$ norm, wikipedia gives us that "The mathematical definition of the $\ell_0$ norm was established by Banach's Theory of Linear Operations. The space of sequences ...
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The meaning of “order of congruence” of metric space

I was studying low-distortion embedding of finite metric space, and was confused about the following concept: Order of congruence: A metric space $(X,D)$ has order of congruence at most $m$ if every ...
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Does there exist a metric $d$ on $\mathbb R$ such that the map $f:(\mathbb R,d) \to (\mathbb R,d)$ ; $f(x)=-x$ is not continuous?

Does there exist a metric $d$ on $\mathbb R$ such that the map $f:(\mathbb R,d) \to (\mathbb R,d)$ defined as $f(x)=-x$ is not continuous?
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subset of a topological space is closed if and only if it contains all of its limit points.

I'm trying to prove the following: Show that a subset of a topological space is closed if and only if it contains all of its limit points. Is my proof valid? Definition of limit point: $p$...
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28 views

If union and intersection of two subsets are connected, the subsets are connected

I've been able to prove what is proved here If union and intersection of two subsets are connected, are the subsets connected? However, I was wondering if I could get some help finding an example to ...
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Show that $\max{\{|a|+|b|,|c|+|d|\}} \leq \max{\{|a|,|c|\}}+\max{\{|b|,|d|\}}.$

Show that $\max{\{|a|+|b|,|c|+|d|\}} \leq \max{\{|a|,|c|\}}+\max{\{|b|,|d|\}}.$ I wanted to show that $d(p,q)=\max{\{|x_1-x_2|,|y_1-y_2|\}}$ where $p=(x_1,y_1),q=(x_2,y_2)$ is a metric on $\mathbb{R^...
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Show that d generates the discrete topology

Let X be any set, and let d be the discrete metric on X. Show that d generates the discrete topology. I just want know if my following proof is valid or not: The discrete topology is the power ...
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17 views

Proof that Euclidean and Uniform norm generate the same topology

I'm teaching myself topology using a book I found. One of the exercises are to prove the following: Define a metric $d'$ on $ \mathbb{R}^n $ ny $d'(x,y) =max${$|x_1-y_1|,...,|x_n-y_n|$}. Show that ...
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Compact Sets of $(X,d)$ with discrete metric

Let $X \neq \emptyset$. Define the discrete metric on $X$ with: $ d(x,y)=\left\{\begin{array}{ll} 1, & x \neq y \\ 0, & x=y\end{array}\right.$ (a) Ascertain the compact ...
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A metric space is compact iff every closed ball is compact

Is this true? I think I have a counter example. If we consider the set $(\mathbb{N},d)$, where $d$ is the discrete metric, then every subset closed ball is compact, but since $\mathbb{N}$ is infinite $...
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47 views

Closure of a set is not necessarily compact

Let $(X,d)$ be a metric space and suppose $(x_n)$ is a Cauchy sequence in $(X,d)$. Is $\overline{\{x_n : n \in \mathbb{N} \}}$ necessarily compact? The answer is obviously no, consider $x_n = 1/n$ ...
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Showing $f$ has a maximum [closed]

Assume that $(X,d)$ is a metric space and that $f: X \to [0, \infty)$ is a continuous function. Assume that for each $\epsilon >0$ there is a compact set $K_{\epsilon} \subset X$ such that $f(x) &...
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50 views

Why do we need $X$ to be compact in the definition of the topology of uniform convergence?

If $(Y,\rho)$ is a metric space and $X$ is compact, then the set $C(X,Y)$ is equipped with a metric $\mu$ thus: $$\mu(f_1, f_2) = \sup_{x \in X} \rho(f_1(x), f_2(x)), f_1, f_2 \in C(X,Y)$$ ...
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17 views

Showing $(C[0,1],d_{\infty})$ is connected

Prove that the metric space $(C[0,1], d_{\infty})$ is connected. Is it path connected? I know how to typically show that a set is connected, but to show $C[0,1]$ is connected is currently escaping ...
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25 views

Holder continuous but not Lipschitz

Is there a function that is Holder continuous but not Lipschitz continuous?
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67 views

Showing the continuity of $d(x,f(x))$

Assume that $(X,d)$ is compact, and that $f: X \to X$ is continuous. Show that the function $g(x) = d(x,f(x))$ is continuous and has a minimum point. Consider the function $g(x) = d(x,f(x))$. If $g$ ...
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34 views

Showing this set is not path connected

Show that the region $\{(x,y) \in \mathbb{R}^2 : x< 0 \ \text{or} \ x>1 \}$ is not path connected. Suppose that $\{(x,y) \in \mathbb{R}^2 : x< 0 \ \text{or} \ x>1 \}$ is path connected. ...
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24 views

$X$ and $Y$ are Compact Metric Spaces Such that for a continuous surjective $f$, $f^{-1}[\{y\}]$ is connected

Let $X$ and $Y$ be compact metric spaces and let $f: X \rightarrow Y$ be a continuous onto map with the property that $f^{-1}[\{y\}]$ is connected for every $y \in Y$. Show that if $Y$ is connected ...
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Image of a disconnected set is disconnected

I'm aware that the image of a connected set is connected and the preimage of a disconnected set is disconnected. However, I'm struggling to find an example of a disconnected set such that the image of ...
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16 views

A function taking values in a countable product of metric spaces is uniformly continuous iff its coordinate functions are.

Let $\lbrace (M_n,d_n) \rbrace _{n \in I}$ be a countable family of metric spaces and define a metric in the product $\prod _{n \in I}M_n$ as follows: $$d(\vec{x},\vec{y})=\sum _{n \in I} \min \lbrace ...
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Show: $(X,d_X)$ is complete $\Leftrightarrow $ $f(X)$ is closed in $(Y, d_Y)$ ($f: X \to Y$ is an isometric embedding)

I have the following task: Show that a metric Space $(X,d_X)$ is complete if and only if for every isometric embedding $f: X \to Y$ in another metric Space $(Y, d_Y)$ it holds true that $f(X)$ is ...
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Is my proof/thought regarding metric spaces and mappings correct? [closed]

Let $(X,d)$ be a metric space $A,B \subset X$ and $A\cap B = \varnothing$ and $f:X \rightarrow \mathbb R $ continuously. Show $0 \le f \le 1$ and $f^{-1}(0)=A$ and $f^{-1}(1)=B$. $$f(x)= \frac{d(x,A)...
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Prove that $d$ and $d'$ generate the same topology on $M$.

I'm teaching myself topology using a book I found. One of the exercises are to prove the following: Let $(M,d)$ be a metric space, let $c$ be a positive real number, and define a new metric $d'$ ...
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Uniform convergence for particular values of $p$

The functions $f_n$ on $[0,1]$ are given by $$f_n(x) = \frac{nx}{1+n^2x^p} \ \ (p >0).$$ For what values of $p$ does the sequence converge uniformly to its pointwise limit $f$? Consider that \...
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Motivation and intuition behind concept of totally bounded-ness in metric spaces

The way I understand it is, a set in a m.s. is totally bounded means the set Admits a finite open cover of fixed size. Regardless of whether it admits an arbitrary open cover. Why do we need totally ...
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Show that $(M, \xi)$ is a complete metric space.

Let $(X,d)$ and $(Y,\rho)$ be metric spaces, and let $M = X \times Y$. Define $\xi : M \times M \to \mathbb{R}$ by $$\xi((x_1, y_1), (x_2,y_2)) = d(x_1,y_1) + \rho(x_2,y_2).$$ Prove that $(M, \xi)$ is ...
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For what $m,n$ does the limit exist.

Let $f: (0, \infty) \times (0, \infty) \to [0, \infty)$ by given by $$f(x,y) = \frac{x^ny^m}{x+y}.$$ Find all $m,n$ such that $\lim_{(x,y) \to (0,0)} f(x,y)$ exists. Consider the line $y=kx$ for ...
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Complete metric space question [closed]

I am stuck on this question. I think that the answer for both (i) and (ii) should be the same, since alpha is continuous on $[0,1]$, so bounded.
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Isometric isomorphism between $R^2$ and $R$

Can someone help me solving the following problems? $(\mathbb R^2,d_2)$ and $(\mathbb R, d_1)$, $d_2, d_1$ being the respective euclidean norms, are not isometric isomorphic, i.e. there is no ...
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To prove that if a function $f : \mathbb R \to \mathbb R$ is locally increasing at every point then the function is increasing [duplicate]

I am trying to prove that if $f : \mathbb R \to \mathbb R$ is locally increasing at every point , i.e. if for every $x\in \mathbb R , \exists r_x >0$ such that $f(a)\ge f(b) , $ whenever $ x+r_x &...
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Hausdorff metric is an Ultra metric

Anybody prove the following statement. Hausdorff metric $H$ is ultrametric if $d$ is ultrametric. For any $A$ and $B$ closed and bounded subsets of $X$, $$H(A,B) = \max \{\sup \limits _{a \in A}...
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30 views

Does a metric Lindelöf space have a countable basis?

I want to prove that every metric space which is Lindelöf has a countable basis. First I tried to show that a countable cover, which exists by the Lindelöf property, is a countable basis, but for the ...
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Does a separable metric space have a countable basis? [duplicate]

I want to prove that if $X$ is a metric space and has a dense countable subset, then it has a countable basis. I know that every metric space is first countable, but I can't continue. Thanks for your ...
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$d_1, d_2$ are metrics on $X$; $(X,d_1)$ is complete. Let $i:(X,d_1)\to(X,d_2)$ be continuous and $i^{-1}$ unif. cont. Show $(X,d_2)$ is complete.

This is a problem on an old preliminary exam in Analysis I'm working through. The problem initially looked easy to me; my plan is to show that for any $\{x_n\}$ Cauchy in $(X,d_2)$, we have that $\{i^{...
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Hausdorff metric, Ultra metric

Can anyone prove the following statement though it seems simple. Let $(X,d)$ be an ultrametric space and $A$ and $B$ are closed, bounded subsets of $X$. Then for each $a$ in $A$ and $\varepsilon > ...
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A subset $X \subset \mathbb{R}^n$ is connected means $X\backslash\lbrace z \rbrace$ is connected

I know that this is true for any $z \in X$, but I am unsure of how to prove it. I was thinking that the best approach would be to demonstrate that $X\backslash{z}$ is path connected, but I'm unsure ...
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Finding similarity of strings using distance function : Bounding the distance function?

I want to know if 2 binary strings $s$ and $t$ each of $d$ length (dimension) and N = 2 (the alphebet) in this case 0 and 1 are similar to each other or not using the following distance function where ...
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26 views

Ambiguous intersection notation: $\mathcal G(\mathbb R^{[0,1]})\cap C([0,1])$

I am attempting to understand a theorem from my lecture's notes, however it seems to me that the notation used there is ambiguous and therefore unclear. The full statement and proof: Where $\...
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Shift of first category set in a compact metric space.

Question. If $X$ is a homogeneous compact metric space, and $F=\bigcup _{n\in\omega}F_n$ is a countable union of closed nowhere dense subsets of $X$, then is there a homeomorphism $\varphi:X\to X$ ...
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Type / Notation of a distance in metric spaces

I've been reading a book and I stumbled upon a notation that I don't understand. Let $(X, d)$ be a metric space. If $A$ is a partition of $X$, then we may consider the metric space $(A, d|_{A \times A}...
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Show at least one limit point

Show that if $r\in\mathbb{R}\backslash \mathbb{Q}$, then $\{e^{i2\pi r n}\}_{n\in\mathbb{N}}$ have at least one limit point I've been sitting with this problem for at while now, but can't figure it ...
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14 views

How to turn a NON-strict total order into a strict total order with $R^3$ vectors?

I'm currently working with colored images in the RGB color space. It's trivial to find a ordering in grayscale images (each shade of gray can be though as a value and darker shades comes before than ...
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108 views

Definition of Cauchy Sequence

I have a question regarding the definition of a Cauchy sequence of a sequence in a metric space. The definition I learned and that is consistent with Wikipedia defines a sequence $(x_n)_{n=1}^\infty$ ...
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1answer
22 views

Prove there exists an open set containing a closed set disjoint with another closed set

Let $F$ and $G$ be closed sets in a metric space $X$ and $F \cap G = \emptyset$. Show that there exists an open set $U$ such that $F \subseteq U$ and $\bar{U} \cap G = \emptyset$. I tried proving ...
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Norms That Define An Open Set

How can a norm define a set in a vector space. I don't understand for example how 2 different norms can define a same open set. It's not intuitive to me. An open set doesn't need a norm to be open (...