# Tagged Questions

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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### 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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### 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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### 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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### Hausdorff metric is an Ultra metric [duplicate]

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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### 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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### 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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### 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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### 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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### 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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### 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 (...
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### Problem regarding isometric isomorphisms [duplicate]

I need help regarding the following two exercises: a) Show that $(\mathbb R^2, d_2)$ and $(\mathbb R,d_1)$ $d_2,d_1$ being the respective euclidean metrics, are not isometric isomorphic, i.e. ...
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### Non-convergent Cauchy sequence in $\ell^1$ with respect to the $\ell^2$ norm

Let $X = \ell^1$, the set of absolutely convergent real valued sequences and let $d_2(\mathbf{x},\mathbf{y}) = \left(\sum_{k=1}^\infty |x_k - y_k|^2\right)^{1/2}.$ This is the $2$-norm on the $1$ ...
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### Showing two norms is not equivalent

Define the norms as $||f||_u=sup_{x\in[a,b]}\{|f(x)|\}\ \ \ \ \ \ \ ||f||=||f||_u+||f'||_u$ Show that $||\cdot||$ and $||\cdot||_u$ is not equivalent I've found a sequence for which $||\cdot||_u$ ...
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### Metric on the profinite completion of the integers?

The p-adic integers come with a metric and associated topology, both of which can be restricted down to the integers. Does this also apply to the profinite completion of the integers? Do they have ...
Let $d_1=|x-y|$ and $d_2=|φ(x)-φ(y)|$ where $φ(x)=\frac{x}{(1+|x|)}$ I must proof $d_1$ and $d_2$ define the same topology over $R^2$ I want some hint. just some indication or méthodes .
### Does $f_n(x)=\sqrt{x-a+1/n}$ converge in $C^1$?
Define $f_n(x)=\sqrt{x-a+1/n}$ on $[a,b]\in\mathbb{R}$. Correct me if I'm wrong, but I worked out that: 1) Since $x\geq a$, then $x-a$ is positive (or $0$), and the function is defined for every \$...