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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Limit points of a subset.

Let $X$ be a metric space, and let $E$ and $K$ be two sets such that $E\subset K$. I want to prove: If $p$ is a limit point of $E$, then $p$ is a limit point of $K$. Proof: if every neighborhood of ...
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7 views

Part of proof of contraction mapping theorem, inequality.

The following is part of a proof I am considering for the contraction mapping theorem. I am not having problems with any part of understanding what is going on with the theorem, or what is happening ...
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1answer
38 views

The set of all condensation points is perfect

Define a point $p$ in a metric space $X$ to be a condensation point of a set $E\subset X$ if every neighborhood of $p$ contains uncountably many points of $E$. Suppose $E\subset \mathbb{R}^k$, $E$ is ...
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1answer
27 views

Open sets relative to a metric space

Let $X$ be a metric space. Let $E\subset Y\subset X$. By an example, it is possible to find a set $E$ which is open relative to $Y$, but not open relative to $X$. The classic one is the segment ...
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1answer
34 views

What functional space does $\mathbb{X} = \{0\}$ belong to?

In a lot of proofs regarding spaces, the example $\mathbb{X} = \{0\}$ is given as the trivial case. Why is that $\mathbb{X} = \{0\}$ is a linear/normed/Banach/Hilbert... space when it is essentially ...
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Equivalent Criteria for convergence of a sequence

My textbook of Metric Spaces describes the following equivalent criteria for convergence of a sequence $\{x_n\}$ : $(i)~ \bigcap \{ \overline {x_n~|~n \in S}~|~S \subseteq \mathbb N, S$ infinite $\} ...
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28 views

Open and closed sets in an arbitrary metric space example

I have the following point lifted directly from my notes on open and closed sets in arbitrary metric spaces: "As an example of a set that is not open, consider $I=[0,1]\subset\mathbb R$, with the ...
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Show that $\mathcal{G}$ generates $\mathcal{B}_X$, the Borel $\sigma$-algebra of $X$.

Let $X$ be a second countable metrizable space and $\mathcal{G}$ a subbase for the topology of $X$. Show that $\mathcal{G}$ generates $\mathcal{B}_X$, the Borel $\sigma$-algebra of $X$. Also show that ...
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Proof that the closed interval in $\mathbb{R}$ is connected

Let $C$ be an open and closed subset of $[a,b]$. WLOG, assume $a \in C$. Set $A = \{x \in [a,b]: [a,x] \subseteq C\}$. Since $a \in A$, sup$A$ exists. Let $\epsilon > 0$. Then, (from real ...
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1answer
36 views

Showing that $d_\infty(x,y)$ is finite for all $x,y\in\mathscr L^\infty$ [closed]

I want to show that $d_\infty(x,y)$ is finite for all $x,y\in\mathscr L^\infty$, however I am not to sure how to go about doing so. We have, $$d_\infty(x,y)=\sup_i|x_i-y_i|$$ and $\mathscr ...
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1answer
29 views

Does this metric induce the topology on the product space?

Consider $\left\{1,2,3\right\}$ with the discrete topology on it. Moreover, consider $Z=\left\{0,1,2\right\}^{\mathbb{Z}}$ with the associated product topology. The cylindersets $$ ...
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Metric spaces, manipulating the absolute value function.

I have the following problem involving the set $Y$ of infinite sequences that absolutely converge such that, $$\sum_{i=0}^\infty x_i^2 \lt\infty$$ where $x_i$ is the $i$-th term in the infinite ...
5
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1answer
47 views

Problem in Banach Fixed Point Theorem for a functional equation

I was recently presented this within the context of topological spaces: I am asked to show that there exists a unique continuous function $ f\colon \left[0,\frac{1}{2}\right] \rightarrow \Bbb R $ ...
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1answer
26 views

Manipulating the maximum function, metric spaces.

I am trying to show that the supremum metric, $d_{\infty}$, is indeed a metric on $\mathbb R^2$. I have shown that the first two properties of a metric space hold, but am having trouble showing the ...
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1answer
36 views

Big an open ball inside small open ball in metric space

When i was reading a book "Elements of the Theory of Functions and Functional Analysis " ([A. N. Kolmogorov, S. V. Fomin) I encountered with very interesting(for me) problem. Problem: Creat a ...
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43 views

On any continuous map $f:S^1 \to \mathbb R$

Let $f:S^1 \to \mathbb R$ be any continuous map , where $S^1$ is the unit circle in the plane . Let $A:=\{(x,y) \in S^1 \times S^1 : x \ne y , f(x)=f(y)\}$ ; then how to prove $A$ is uncountable , or ...
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3answers
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If $E$ and $F$ are disjoint closed subsets of a metric space $(X,d)$, then is $dist~(E,F) >0$ always? [duplicate]

If $E$ and $F$ are disjoint closed subsets of a metric space $(X,d)$, then is $dist~(E,F) >0?$ My attempt: Suppose $dist~(E,F)=0.$ Then $\exists~e \in E,f \in F$ such that $~\forall ...
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1answer
67 views

Norm $\Vert \cdot \Vert$ on the symmetric group $S_n$

If we define a real valued function $\Vert \cdot \Vert$ on the $n^{th}$ order symmetric group $S_n$ satisfying following conditions $$\begin{align} & \|x\|=0\iff x=\omega\,\,\,(\text{identity ...
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1answer
28 views

Is the closure of every bounded convex set , with non-empty interior , in $\mathbb R^n (n>1)$ homeomorphic to a closed ball?

Is the closure of every bounded convex set , with non-empty interior , in $\mathbb R^n (n>1)$ homeomorphic to a closed ball (by closed ball I mean $B[a,r]:=\{x \in \mathbb R^n : d(x,a)\le r\}$ , ...
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1answer
43 views

How is this metric defined when there is no $k$?

Let $\left\{1,2,3\right\}$ be equipped with the discrete topology and $X=\left\{1,2,3\right\}^{\mathbb{Z}}$ with the product-topology. Then one possible metric on $X$ is $$ d(x,y)=\begin{cases}2^{-k} ...
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Is the inverse of a bijective connectedness preserving map , on a complete real inner product space , also connectedness preserving?

Let $X$ be a complete real inner-product space and $f:X \to X$ be a bijection which maps connected sets to connected sets ; then is it necessarily true that $f^{-1}$ also maps connected sets to ...
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Spaces whose all their metrizations are complete [duplicate]

Which metrizable topological spaces $(X,\tau)$ posses the following property: Every compatible metric (i.e one which induces the same topology $\tau$) is complete. Compact metrizable spaces satisfy ...
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60 views

$\mathbb{N}$- a complete metric space with $d(x,y)=|x-y|$

$\mathbb{N}$- a complete metric space with $d(x,y)=|x-y|.$ This seems quite intuitively correct, but I do not know how to prove this formally, does anyone know how they would go about this?
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What are some functions $f \in L^\infty(\Omega)$

In my text book all it said about $L^\infty(\Omega)$ space is that it is the space of all measurable functions that are bounded almost everywhere. No example given. I can't see how any member of this ...
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1answer
26 views

Which subsets of a given real linear space $V$ are open unit balls with respect to some norm on the space?

Which subsets of a given real linear space $V$ are open unit balls with respect to some norm on the space? My textbook (Metric Spaces - Michael Searcoid) gives the following hints to answer the above ...
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1answer
29 views

Is there a name for embedding of Lebesgue spaces $L^q(a,b) ⊂ L^p(a,b)$

Let $1 ≤ p ≤ q ≤ ∞$ then $L^q(a,b) ⊂ L^p(a,b)$ where $L^q(a,b), L^p(a,b)$ are Lebesgue spaces Is there a name for this relationship? Is this the Sobolev embedding theorem?
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How to prove $f:X\to Y$ is continuous

$X,Y$ are metric spaces. Then $f:X\to Y$ is continuous in $X$ if that $C\subset Y$ is closed implies that its inverse image is closed in $X$. I want a proof that's directly based on the ...
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1answer
28 views

Are Homogenous countable complete metric spaces always discrete?

Let $M$ be a countable complete metric space such that the group of isometries of $M$, $Iso(M)$ acts transitively on the points in $M$. Does it follow that the topology induced by the metric is the ...
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1answer
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Bounded sets equivalent definition

Let $X$ be a metric space, and $E\subset X$. I have two definitions of a bounded set, I want to prove they are equivalent. Definition 1: $\exists M:\exists q\in X:\forall p\in E:d(p,q)<M$ ...
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Proving that $S^c=\left\{f(x)\in C^2[0,1]\;\Big\vert\; \int_{0}^{1}f(x)dx > 3\right\}$ is open in $C^2[0,1]$ with a specific metric

I am trying to prove that $$S^c=\left\{f(x)\in C^2[0,1]\;\Big\vert\; \int_{0}^{1}f(x)dx > 3\right\}$$ is open in $C^2[0,1]$ with the metric $d$ given by $$ d(f,g)=\sup_{x \in [0,1]}|f(x)-g(x)|+ ...
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1answer
36 views

Definition of standard deviation and $l_2$

If we denote the mean as $\mu$, then the standard deviation is: $$\sigma\equiv\left(\sum_{x\in X}{p(x)(x-\mu)^2}\right)^\frac{1}{2}$$ In other words, $\sigma$ is the average $l_2$ distance from $\mu$. ...
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1answer
46 views

Completeness of ${C^2[0,1]}$ with under a specific metric

Prove that ${C^2[0,1]} $ (set of two times differentiable functions)is complete with metric: $$d(f,g)=\sup_{x \in [0,1]}|f(x)-g(x)|+ \sup_{x \in [0,1]}|f'(x)-g'(x)| + \sup_{x \in ...
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Prove that Euclidean distance in $\mathbb{R}^n$ is a distance

I'm trying to show that: $$\forall x,y\in\mathbb{R}^n, d(x,y)=\left(\sum_{i=1}^n(x_i-y_i)^2\right)^{1/2}$$ is a distance. However I have not proved Cauchy-Schwarz yet and I'm pretty sure I wouldn't ...
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1answer
41 views

Problem in showing that a sequence is a Cauchy sequence on a space with the integral metric.

I'm having difficulty following what is going on and understanding parts in the following example. It is quite similar to an example I posted before (Changing of the limits of integration with the ...
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2answers
46 views

Why is the topology on $\mathbb{R}$ formed by the basis $[a,b)$ normal?

Why is the topology on $\mathbb{R}$ formed by the basis $[a,b)$ normal? I need to prove that two disjoint closed sets are contained wtihin two open disjoint sets. First, I tried to understand how a ...
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Defining a metric in the tangent spaces $ T_xM $

I'm working with a metric $D$ over the manifold of grassman $G_n(\mathbb{R}^{d})$ and have difficulties to extend $D$. Let me explain: If $M$ is a submanifold of dimension $n$ in $\mathbb{R}^{d}$ ...
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4answers
101 views

Are there an infinite number of open balls in an open set in a metric space?

Let's start off by recalling the definition of an open set in a metric space: A set $A$ in a metric space $(X,d)$ is open if for each point $x\in A$ there is a number $r\gt0$ such that $B_r(x)\subset ...
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1answer
43 views

The convergence of different metrics on the same space

The following example is from my notes, and I would like clarification on some wider points connected to it, namely about extensions from what we understand metrics and metric spaces to be. It follows ...
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1answer
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Changing of the limits of integration with the integral metric.

Consider the following sequence of functions, $$f_n(x) = \begin{cases} nx & \text{for $0\le x \le \frac1n$} \\ 1 & \text{for $x\ge \frac1n$} \end{cases}$$ And call to mind the integral ...
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On a metric over m-subsets of [n]

Given an integer $n$, denote the set of integers $\{1,2,\dots,n\}$ as $[n]$. For two $m$-subsets $A$ and $B$ of $[n]$, list their elements in the increasing order as $a_1 < a_2 < \dots < a_m$ ...
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1answer
40 views

Showing $d(x,y)=0$ iff $x_{n}=y_{n}$

Consider the space $\mathbb{R}^{\infty}$ of all sequences $x=\left \{ x_{1},x_{2},... \right \}$ of real numbers. Define the function $d:\mathbb{R}^{\infty}\times \mathbb{R}^{\infty}\rightarrow ...
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2answers
40 views

Predicate logic inference in a simple proof of uniform continuity.

For a function $f$ from a metric space $X$ into a metric space $Y$, uniform continuity can defined in this way: $\forall ε>0:\existsδ > 0:\forall p,q\in X:d_{X}(p,q)<δ \rightarrow ...
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3answers
13 views

$A$ be a convex subset and $D$ be dense in a real NLS ; then for every non-empty open subset $U$ of the NLS , $U\cap D\cap V \ne \phi$?

Let $A$ be a convex subset of a real normed linear space $V$ and $D$ be a dense subset of $V$ ; then is it true that for every non-empty open subset $U$ of $V$ , $U\cap D\cap A $ is non-empty ?
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Complement of the union of countably many , mutually disjoint , non-empty open balls in $\mathbb R^n , (n >1) $ is path connected?

Let $n \ge 2$ and $\{B_m\}_{m=1}^\infty$ be countably infinitely many , mutually disjoint , non-empty open balls in $\mathbb R^n$ , then is $\mathbb R^n \setminus \cup_{m=1}^\infty B_m$ ...
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1answer
55 views

Confused about the open/closed set in metric space

Let $(M,d)$ be a metric space. I understand well that $\emptyset$ and $\mathbb{R}$ are both open and closed sets. I read some notes that say, that $\emptyset$ and $M$ are both open and closed. So, ...
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1answer
50 views

Distance between a point and an empty set: meaning and value?

On page 253 in General Topology by R Engelking: The distance $\rho(x, A)$ from a point $x$ to a set $A$ in a metric space $(X,\rho)$ is defined by letting $\rho(x, A) = \text {inf}\ {\{\rho(x, a) ...
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0answers
31 views

Statement about the discrete (metric) space, and both an open and closed ball.

I have the following statement from my notes: "Let $(X,d)$ be the discrete space i.e. any non-empty set with the discrete metric ($d_d(x,y)=1$ for all $x\neq y$). Then, amazingly, $B_1(x)=\{x\}$, a ...
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0answers
16 views

Find Connected Components

Let $X = \mathbb{R^2}$ and let $F_k$ be the closed line segment joining $(-1,2^{-k})$ and $(1,2^{-k})$ Additionally let $a = (-1,0) \ ,\ b=(1,0)$ Consider $A = \{a,b\} \cup \ ...
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1answer
38 views

Dense $G_{\delta}$ set implies comeagre set

Suppose that $X$ is a metric space. Show that if $D$ is a dense $G_{\delta}$ set, then $D$ is comeagre, that is, countable intersection of dense sets. My attempt: Let $D=\bigcap_{n \in ...
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1answer
64 views

Why is the Gromov-Hausdorff distance a metric?

The Gromov-Hausdorff distance is: $$ d_{GH}(A,B) = \inf_{f,g}d_H(A',B') $$where $f$ and $g$ are isometric embeddings of $A,B$ into some metric space, and their images are $A', B'$. The inf is taken ...