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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convergence in metric space
Let $C[-1, 1]$ be the space of continuous functions equipped with the metric $(f, g) = \displaystyle\max_{x \in [-1, 1]} |f(x)-g(x)|$.
Consider the sequence $(f_n)$ of functions $f_n : [-1, 1] \to ...
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43 views
Determining Complete Metric Spaces
I need to determine if $((0, 1), d)$ where $d(x, y) = |x^2 - y^2| \forall_{x,y}\in (0,1)$
My argument is as follows, take the sequence defined by $\dfrac{1}{n}$, then we know by $d(x, y)$ that ...
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1answer
31 views
Correctness of Analysis argument with Cauchy sequences
Let $(x_n)$ and $(y_n)$ be Cauchy sequences in $(X, d)$. Show that $(x_n)$ and $(y_n)$ converge to the same limit iff $d(x_n, y_n) \rightarrow 0$
Proof $\rightarrow$
Suppose $(x_n) \to a$ and $(y_n) ...
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1answer
25 views
Extension on pseudometric spaces using uniform continuity
I would like to know if the extension theorem of uniformly continuous
functions can be generalized to pseudometric spaces.
That is, let $X,Y$ be a pseudometric spaces and $D\subset X$ a dense ...
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1answer
35 views
Cauchy Sequences and Analysis
Let $(x_n)$ and $(y_n)$ be Cauchy sequences in a metric space $(X, d)$. Show that the sequence $(d(x_n, y_n))$ is a cauchy sequence in $\mathbb{R}$.
What is the significance of $\mathbb{R}$ in this ...
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1answer
60 views
Cauchy in metric space
Let $C[-1,1]$ be the space of continuous functions with metric $$\rho(f,g)=\max\{|f(x)-g(x)|: x\in [-1,1]\}\;.$$
Then the sequence of functions $(f_n) :[-1,1] \to \Bbb R$ defined as ...
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2answers
47 views
contraction metric space
"Let $0 < a < 1$ and $f(x) = (x^2 + a)/2$. Show that $f : [0, a] \to [0, a]$ and that f is a
contraction. Find the fixed point of $f$."
For this problem, since there isn't any metric $d(x,y)$ ...
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34 views
Distance between a closed and a compact sets. [duplicate]
Let $X$ and $Y$ be non empty subsets of $\mathbb{C}$, $X$ closed and $Y$ compact and $X\cap Y=\emptyset$.
Define: $d:=\sup \{r:(\forall x\in X, y\in Y) \vert x-y \vert \geq r \}$.
Show: ...
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1answer
48 views
What values of $p$ make $d$ a metric?
I'm trying to determine what values of p make the following a metric:
$$d(x,y)=|x-y|^p$$ for x,y∈R.
Obviously, it's not difficult to show that this satisfies the first two conditions for most values ...
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1answer
60 views
determine whether a metric space is complete or not
How to decide if the metric spaces $((0,1)$, $d(x,y)=|x^2-y^2|)$ and $((-\frac{\pi}{2},\frac{\pi}{2})$, $d(x,y)=|\tan x-\tan y|)$ are complete or not.
For the first metric, I let any cauchy sequence ...
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2answers
47 views
Would this be a metric?
I just read about the taxicab metric defined on $\mathbb{R}^2$.
Suppose you have the plane with fixed $x$-axes and $y$-axes. A path from point A and B on the plane must satisfy: you can only move in ...
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2answers
22 views
Equality of limits with respect to different metrics.
Suppose that $X$ is a set equipped with two metrics, say $d_1$ and $d_2$. Let $\{x_n\}_{n\in\mathbb{N}}\subset X$ be a sequence of points which converges to $x\in X$ with respect to metric $d_1$. ...
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4answers
98 views
Problem of Separable Metric Space, Isolated points and countable sets
How to prove that if a metric space $(E, d)$ is separable and $A\subseteq E$ is a set where all points are isolated then A is countable.
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2answers
104 views
Show that the countable product of metric spaces is metrizable
Given a countable collection of metric spaces $\{(X_n,\rho_n)\}_{n=1}^{\infty}$. Form the Cartesian Product of these sets $X=\displaystyle\prod_{n=1}^{\infty}X_n$, and define $\rho:X\times ...
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88 views
Showing the metric $\rho=\frac{d}{d+1} $ induces the same toplogy as $d$
I want to show that given a metric space $\left(X,d\right)$ the metric $\rho=\frac{d}{d+1}$ on $X$ induces the same topology on $X$ as $d$. It suffices to show that any open ball in ...
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39 views
Intuitive explanation of ball-based definition for continuity of functions in metric spaces
First of all, hat tip to @Fayz for providing this definition.
Backstory: I broke my glasses several days ago and, in the meantime, this important definition was written on a board I could not see. ...
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1answer
41 views
Function doesn't increase distance
Let $(X, d)$ be a metric space, $A \subseteq X$ with $A \neq \varnothing$, and $$f: A \rightarrow \mathbb{R}\quad\text{such that}\quad \left|f(x)-f(y)\right|\leq d(x,y),\ x,y\in A.\tag{$\ast$}$$
Let ...
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3answers
28 views
what would be a example in a general metric space about closedness and boundedness not imply sequential compactness?
Unlike in $R^n$, closed and bounded doesn't guarantee sequential compactness. Textbook examples includes sup metric and R^infinite metric. I am wondering what would be a example of closed and bounded ...
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43 views
Need to confirm: Sup Metric $C[0,1]$, question about boundary
For the sup metric, $C[0,1]$. Let $S \subset C[0,1]$ be given by:
$$S=\left\{f:[0,1]\to \mathbb{R} \ : \ 0 \leq f\left(\frac{1}{2}\right)<1\right\}$$
The question is simple: is this set open or ...
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2answers
32 views
covering positively disjoint sets with finite balls
I have a question if $X$ is a separable bounded metric space and $A$,$B\subset X$ are sets such that
$d(A,B)>0.$
Does there exists sets $A^{\prime}$ and $B^{\prime}$ such that each of them is
a ...
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1answer
75 views
Continuity in metric space, TRUE or FALSE?
Let $(X,d)$ and $(Y,e)$ be metric spaces , and let $f: X \to Y$ be a function.
True or false ? Give a proof or a counterexample as appropriate.
$(a)$ If $d$ is the discrete metric on ...
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2answers
49 views
Group, metric, completion
Let $G$ be a group, $(G, \rho)$ - metric space, $p: G \rightarrow \mathbb{R}_+$ such that $p(x)=0 \iff x=e_G, \ \ p(x^{-1})=p(x), \ \ p(xy)\le p(x)+p(y), \ \ p(xy)=p(yx)$
Now let $\rho ...
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1answer
37 views
Function which doesn't increase distance
Let $(X, d)$ be a metric space, $A \subseteq X$ with $A \neq \varnothing$, and $$f: A \rightarrow \mathbb{R}\quad\text{such that}\quad \left|f(x)-f(y)\right|\leq d(x,y),\ x,y\in A.\tag{$\ast$}$$
Let ...
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1answer
226 views
Prove that if $Z\subseteq Y$, then $(g\circ f)^{-1}(Z)=f^{-1}(g^{-1}(Z)).$
Let $W ,X$ and $Y$ be three sets and let $f :W \to X$ and $g: X \to Y$ be two functions. Consider the composition $g \circ f: W \to Y $ which, as usual , is defined bt $(g\circ f)(w)=g(f(w))$ for ...
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1answer
26 views
Symmetric Operator with Different dot products
If I have a symmetric operator $A$ in a metric space $\mathscr{M}$.
Then $\langle Au,v\rangle =\langle v,Au\rangle $ with the dot product defined in $\mathscr{M}$.
My question is, if I keep the same ...
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3answers
57 views
Continuous map between metric spaces
Suppose $X,Y$ are metric spaces, let $A \subset X$ be a bounded subset of $X$ and $f: A \to Y$ to be a continuous bjection. Prove or disprove that $f^{-1}$ is continuous.
Remark: If each closed ...
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1answer
43 views
Metric problems
I am taking the GRE in less than 10 days, and I have never taken analysis. And I would like to tackle metric problems and I was wondering if anyone could show me a certain strategy to solve problems ...
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1answer
39 views
Let $ f:(X, d) \mapsto (Y,d) $ be an mapping such that $ Graph (f) $ is connected. [duplicate]
Where $ X $ is connected. Does it imply $ f $ to be continuous?
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Regarding nowhere dense subsets and their measure.
A while ago it was made clear that a nowhere dense subset $P \subset [0;1]$ whose Lebesgue measure $\mu(P) = \mu([0;1]) = 1$ doesn't exist.
But is it possible in principle to define a nowhere dense ...
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2answers
58 views
Is the set of integers Cauchy complete?
http://en.wikipedia.org/wiki/Complete_metric_space says that a metric space M is called complete (or a Cauchy space) if every Cauchy sequence of points in M has a limit that is also in M or, ...
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Csiszar divergence for subprobability distributions
I'm wondering whether there is some standard generalization of Csiszar divergences for subprobability distributions and if any classical property of divergences no longer holds in this case.
The ...
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2answers
50 views
Let $ f:(X, d)\mapsto (X, d ) $ be a mapping on compact metric space with $ d (f (x), f (y))<d (x,y) $for $ x\ne y $
I prove that $ f $ has a fixed point. My question is whether the point is unique and the mapping $ f $ is continuous.
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1answer
54 views
How to show that a continuous map on a compact metric space must fix some non-empty set.
Suppose $(X,d)$ is a compact metric space and $f:X\to X$ a continuous map. Show that $f (A)=A$ for some nonempty $A\subseteq X.$
I start this by supposing that $A_0:=X$ and $A_{n+1}:=f(A_n)$ for ...
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2answers
72 views
Prove an inequality in $\mathbb{R}$
Let $ p,q \in \mathbb{R}, \; \lambda > 0, p \neq q$ (two points).
For the two points $x_+, x_{-}$ with
\begin{align*}
x_+&=p+\lambda\cdot (q-p)\\
x_{-}&=p-\lambda \cdot (q-p)
...
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1answer
47 views
Is $(A \times B)^\epsilon \subseteq A^\epsilon \times B^\epsilon$?
While working on a problem related to my research, I had the following query. It pertains to product spaces:
The Question:
Let $(X,d_X)$ and $(Y,d_Y)$ be two Polish (Complete separable metric) ...
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1answer
43 views
Isometry from Manhattan plane to Euclidean plane?
Does there exist an isometry from a Manhattan plane $A$ to a Euclidean plane $B$?
I.e. a function $\varphi:A \to B$ that suffices $\|\varphi(a)\|_B = \|a\|_A$ for all $a \in A$, where $\| \cdot \|_A$ ...
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1answer
30 views
Proving $\ell_\infty$ is complete
I start by taking a Cauchy sequence $(a_i)$ in $\ell_\infty$. I denote the terms of $(a_i)$ as $f_1, f_2, f_3, \dots$ and so on.
For each $x \in \mathbb{N}$, the sequence $(f_1(x), f_2(x), f_3(x), ...
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2answers
72 views
Defining a metric space
I'm studying for actuarial exams, but I always pick up mathematics books because I like to challenge myself and try to learn new branches. Recently I've bought Topology by D. Kahn and am finding it ...
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1answer
48 views
Showing that the open ball is homeomorphic to $\mathbb{R}^n$ [duplicate]
I'm trying to prove that an open ball is homeomorphic to $\mathbb{R}^n$ but since this is the first proof of this kind that I try to give I'm having a little doubt on how to begin. I've had some ...
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126 views
Space of functions that are everywhere differentiable
Define the space $\beta^1([a,b])$ as the space of functions $f : [a,b] \mapsto \Bbb R$ which are everywhere differentiable and whose derivative $f'$ is a bounded function. One equips this space with ...
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6answers
323 views
Why metric space is topological space and examples of non-Hausdorff spaces
I am learning basic topology in my analysis class these days. I have a few questions here:
Why is it true that a metric space is a special form of a topological space?
What are some simple examples ...
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82 views
Is a continuous function like a homomorphism/isomorphism for metric spaces?
If I had to define a notion of a homomorphism/isomorphism on metric spaces, I'd say something like this.
Let $A$ and $B$ be metric spaces with norms $\| \cdot \|_A$ and $\| \cdot \|_B$ respectively. ...
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2answers
54 views
Irrational P-adics
$\mathbb{Q}_p$ is completion of $\mathbb{Q}$ by defining a new metric. So, with respect to this new metric they are complete.
I just want to be sure, are there p-adic rationals? If there are P-adic ...
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1answer
24 views
Prove that $ A\subset \ell_1 $ is compact iff $A$ satisfies the following property
$A$ is compact iff $ A $ is bounded and, given $\epsilon > 0$, there exists $ n_0$ such that $ \sum_ {k=n}^\infty |x_k|\le\epsilon $ for all $n \geq n_0 $ and for all $ x\in A $.
To prove ...
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If $x\in X$ and sequence $(x_n) \in X^{\Bbb N}$ converges in $(X,d)$ , then so does every subsequence of $(x_n)$.
A subequence of a sequence $(x_n)_{n\ge 1}$ is a sequence $ (x_{n_1}, x_{n_2},x_{n_3},...$) where $n_1,n_2,n_3,... \in \Bbb N$ with $n_1\lt n_2\lt n_3\lt ...$
Let $(X,d)$ be a metric space and let ...
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1answer
26 views
Is a 'normally' convergent sequence still convergent in a metric space which barely excludes its 'normal' limit?
For example, suppose
$$ x_n = \frac 1n \\ X = (0, 1)$$
Is $x_n$ convergent in $X$?
My guess would be no, since there exists no $x \in X$ which $x_n$ approaches; $x_n$ will eventually surpass any ...
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1answer
72 views
It is possible to generalize the “real” line to be able to embed $\omega_1$ or any uncountable ordinal into a finite segment of it?
This question is motivated from a previous question, but is in itself independent of it.
So, I understand that it is not possible to embed $\omega_1$ or any uncountable ordinal into the real line, ...
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1answer
38 views
Metric Spaces needed for Differential Geometry
I've asked here about some texts about differential geometry which doesn't assumes that the reader knows general topology. I've got good references as Do Carmo's Differential Geometry of Curves and ...
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1answer
25 views
Equivalents metrics and closed sets
I have proven that if $ d $ and $ \rho $ are two equivalent metrics on a set $ E $ then these metrics define the same open sets in both metric spaces $ (E, d) $ as $ (E, \rho ) $. What I tried was ...
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55 views
Doubt in Spivak's examples of Manifolds
I've started to study Differential Geometry in Spivak's first volume of his Differential Geometry books. I like very much his approach since general topology isn't assumed, and since he gives many ...
