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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Why completion of a metric space $X$ is 'unique' upto isometry?
Let $(X,d)$ be a metric space.
Let $({X_1}^*, {d_1}^*)$ and $({X_2}^*, {d_2}^*)$ be completions of $(X,d)$ such that $\phi_1:X\rightarrow {X_1}^*$ and $\phi_2:X\rightarrow {X_2}^*$ are isometries. ...
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Does a compact subspace have to be closed in an arbitrary metric space?
For Euclidean spaces, we have that a compact subspace has to be closed (and bounded.) But how about an arbitrary metric space? Or how about an arbitrary topology space?
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question regarding metric spaces
let X be the surface of the earth for any two points on the earth surface. let d(a,b) be the least time needed to travel from a to b.is this the metric on X? kindly explain each step and logic, ...
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Two convergent sequences in a metric space.
Question: Let {$x_n$} and {$y_n$} be two convergent sequences in a metric space (E,d). For all $n \in \mathbb{N}$, we defind $z_{2n}=x_n$ and $z_{2n+1}=y_n$. Show that {$z_n$} converges to some $l \in ...
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Convergence in Skorokhod metric and unifrom metric
Is there a relationship between convergence in the Skorokhod space and convergence in the uniform metric. I.e. does weak convergence in the Skorokhod space imply convergence in the uniform metric?
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Prove that the sequence $\{ z_n\}_{n\in \mathbb{N}}$ is cauchy?
I am given the following problem
Let (X; d) be a metric space, and let $\{ y_n\}_{n\in \mathbb{N}}$ and $\{ x_n\}_{n\in \mathbb{N}}$ be two
sequences in $(X, d)$, both converging towards $a\in ...
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Is there an associative (monoid) operation on $\mathbb{R}_{\geq 0}$ which is also a metric?
Is there an associative operation $\star \colon \mathbb{R}_{\geq 0} \times \mathbb{R}_{\geq 0} \to \mathbb{R}_{\geq 0} $ wich also happens to be a metric on $\mathbb{R}_{\geq 0}$? Furthermore is there ...
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A problem about topologically equivalent metrics
I tried to solve this problem:
Let $(X,d)$ a metric space. Show that $d$ and $\bar{d} =\min({d(x,y),1})$ are topologically equivalent metrics.
I proved that $\bar{d}$ is a distance, then I tried to ...
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A point in subset $A $of metric space is either limit point or isolated point.
Let $A$ be a subset of a metric space. $A'$ be the set of limit points of $A$ and $A^i$ be the set of isolated points of $A$.
Show $A \subset A' \cup A^i$.
The picture on my mind is that I draw a ...
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Every point in a metric space has at least a neighborhood?
I was reading about topology and I came across this statement:
Every point $x$ in metric space $(X,d)$ has a neighborhood, which a neighborhood of $x$ (denoted $N(x)$) is defined as there exists ...
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Symmetric Difference as a Metric
I'm currently working on a homework question, and I am stuck. I copied this off the whiteboard, so it is very possible that I made a mistake in transcribing it. Of course I may be misinterpreting ...
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Challenge question for normed linear spaces.
I have come across the following challenging problem in my analysis course: Let $K$ be a compact convex set in a normed linear space. Suppose that
$$\sup_{x,y\in K}\{||x-y||\}=\delta>0.$$
Show ...
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what is this called? “difference of the function is less than the function of the difference”
Given:
a metric $d$
an aggregate function $f$
some sets (or multisets or random variables) $X$,$Y$
What do we call:
$d(f(X),f(Y)) \leq f( [d(X_0,Y_0) \cdots d(X_n,Y_n)] )\ \forall\
...
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81 views
Completing $\Bbb R$ when some “divergent” sequences are Cauchy sequences
If I equip $\mathbb{R}$ with the metric
$$
\rho(x,y) := \left|\arctan(x) - \arctan(y)\right|
$$
then sequences like for example $x_n = n$ are Cauchy sequences, so it is clear that $\mathbb{R}$ is ...
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Given a metric d(x,y), what types of functions will produce [closed]
Possible Duplicate:
What operations is a metric closed under?
Say we have a metric $d: X \rightarrow \mathbb{R}$, and a function $f(d)$ that takes in a metric $d$ and (ideally) spits out ...