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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93 views

Mean value of convergent series

Let us in a normed linear space have a sequence $\{a_i\}_{i=1}^\infty$ which converges to some value $b$, how can I show that $$\lim_{n\rightarrow\infty}\sum_{i=1}^n\frac{a_i}{n}=b$$ My idea is to use ...
5
votes
3answers
677 views

Is a finite union of bounded sets bounded in any metrical space?

In any metrical space $(M,d_M)$, consider $n$ bounded subsets $S_i\subset M$. Then, is $\cup_i^nS_i$ bounded? If so, why?
3
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2answers
669 views

Is the complement of a countable set in $\mathbb{R}$ dense? Application to convergence of probability distribution functions.

I am wondering if we have a set $A\in\mathbb{R}$ that is countable, whether $A^{c}$ is dense in $\mathbb{R}$? I thought I saw this quoted somewhere on google but I can not find it again! I am working ...
1
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1answer
377 views

Sets where Heine-Borel theorem works [duplicate]

Possible Duplicate: Are there more general spaces than Euclidean spaces to have the Heine–Borel property? By Heine-Borel theorem, a closed and bounded subset of the Euclidean space is ...
5
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1answer
1k views

Prove the map has a fixed point

Assume $K$ is a compact metric space with metric $\rho$ and $A$ is a map from $K$ to $K$ such that $\rho (Ax,Ay) < \rho(x,y)$ for $x\neq y$. Prove A have a unique fixed point in $K$. The ...
3
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2answers
452 views

$X$ is connected iff $\forall A\subset X,$ $\partial A\neq\emptyset$

Prove metric space $X$ is connected iff $\forall A\subset X,$ $\partial A\neq\emptyset$. Attempt at a proof: $\rightarrow$ $X$ connected $\implies$ $\forall A\subset X$, $A$ is connected. Then, ...
2
votes
1answer
309 views

An interval is path connected

$A$ is an interval $\implies$ $A$ is pathwise connected. This kind of goes off one of my previous general questions about path connectedness. I've tried to formalize my attempt at proving this ...
3
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2answers
613 views

Tori and metrics

I have been doing some reading on tori. What I can make out of it is that a torus can be equipped with different metrics -- locally Euclidean or as an embedded surface. It is said however that the ...
1
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1answer
487 views

connectedness vs. path connectedness

Is there a general rule of what kind of sets it is easier to prove connectedness using path connectedness or regular connectedness? I understand that path connected $\implies$ connected, but are ...
1
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1answer
742 views

How to show $f$ is continuous at $x$ IFF for any sequence ${x_n}$ in $X$ converging to $x$ the sequence $f(x_n)$ converges in $Y$ to $f(x)$

Let $f : (X, d_X) \to (Y, d_Y )$ be a map between two metric spaces. Recall that $f$ is called continuous at $x ∈ X$ if for any open ball $ B_f(x)(ϵ)$ of radius $ϵ$ around $f(x)$ there exists a ball ...
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0answers
128 views

Show that a subspace $X$ of the Euclidean space $\Bbb{R}^n$ is compact if and only if any sequence of elements of $X$ has a converging subsequence.

Remark: this statement holds in the considerably greater generality of any metric space but the proof of this more general result is quite involved.
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1answer
599 views

How to show convergence in a metric space?

Suppose that $\{x_n\}→x$ where $\{x_n\}$ is a sequence in a normed space V and $x ∈ V$. Show that $\forall y ∈ V, \{x_n + y\} → x + y$.
2
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1answer
114 views

Disconnectedness on the Real Line

Theorem. If a set $E \subset \mathbb{R}$ is disconnected, then there exist $x, y \in E$, and some $z \in \mathbb{R} \setminus E$ with $x < z < y$. I want to prove this theorem using the ...
0
votes
1answer
148 views

$A\subset (X,d)$ nowhere dense $\iff$ $(\overline{A})^o=\emptyset$

Show $A\subset (X,d)$ nowhere dense $\iff$ $(\overline{A})^o=\emptyset$. My attempt: $A$ nowhere dense $\implies$ $(\overline{A})^c$ is dense in $X$. Then, $(A^c)^o$ is dense in $X$ (previously ...
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2answers
154 views

Connecting two definitions of connectedness

Let $E$ be a subset of a metric space $X$, and say $E$ is connected if and only if it is not disconnected. I have the following two definitions of disconnected: I) $E$ is disconnected if there ...
4
votes
5answers
2k views

What is the difference between metric spaces and vector spaces?

Does a metric space have an origin? That is, does it have $(0,0)$. Does a vector space have an origin? It seems whatever you can do in a metric space can also be done in a vector space. Is this ...
4
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2answers
120 views

Is Completeness intrinsic to a space?

Is completeness an intrinsic property of a space that is independent of metric? For example, since $\mathbb{R}^n$ is complete with the Euclidean metric, is it complete with any other metric? If ...
2
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1answer
89 views

prove $\mathbb{N}$ is complete w.r.t. $d_2$

Prove $(\mathbb{N},d_2)$ is a complete metric space. Attempt: So I need to show that every Cauchy sequence in this metric space converges. Presumably all of these convergent Cauchy sequences ...
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2answers
172 views

connected components are connected

Show that for $x_o\in X$, the connected component of $x_o$ is connected. Attempt: So I'm trying to show that assuming that the union of connected sets that contain $x_o$ is not connected results ...
2
votes
3answers
212 views

Proving the Cantor Set contains no segment

I know that the Cantor Set contains no segment of the form $$\left(\frac{3k+1}{3^m}, \frac{3k+2}{3^m}\right)$$ for any integers $k$ and $m$. If we can prove that every real segment contains a segment ...
8
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1answer
319 views

Infinite closed partition of the real numbers with a certain property

Is there a partition of the real numbers into infinitely many closed subsets so that no infinite union of these subsets (other than the whole set of real numbers) is closed?
11
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2answers
1k views

if every continuous function attains its maximum then the (metric) space is compact

Suppose $(M,d)$ a metric space. I want to show that if every continuous real-valued function on $M$ attains a maximum, then the space must be compact. I was trying to do this by assuming $M$ ...
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2answers
145 views

Completability of a uniform space, metric space and topological vector space?

From Wikipedia In particular, topological vector spaces are uniform spaces and one can thus talk about completeness, uniform convergence and uniform continuity. (This implies that every ...
1
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1answer
94 views

Lorentz reflection

What is a Lorentz reflection of $\mathbb R^3$? Is there a way to visualize it? Suppose I have a plane, P, what would (Lorentz) reflecting in it differ from (Euclid) reflecting in it? I know that the ...
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2answers
229 views

proving definitions of connectedness are equivalent

Prove these definitions are equivalent: Definition $\,(1)\,$: $A\subset X$ is not connected if for open $U, V\subset X\,\,, $$\,\,U\cap\bar{V}=\emptyset\,\,$, $\,\,\bar{U}\cap V=\emptyset\,\,$, ...
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3answers
196 views

Boundedness in a topological space?

I was wondering if there is a concept of boundedness for subsets of a topological space? If yes to 1, is it this one from Wiki Elements of a Bornology B on a set X are called bounded sets and the ...
1
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4answers
898 views

an open ball in $\mathbb{R^n}$ is connected

Show that an open ball in $\mathbb{R^n}$ is a connected set. Attempt at a Proof: Let $r>0$ and $x_o\in\mathbb{R^n}$. Suppose $B_r(x_o)$ is not connected. Then, there exist $U,V$ open in ...
0
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2answers
107 views

Finding a continuous function with specified properties

This is a homework question in my analysis class: Let $A$ and $B$ be two nonempty closed subsets of a metric space $X$ that do no intersect. Show that there is a continuous function $f:X\rightarrow ...
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5answers
915 views

Set $A$ interval in $\mathbb{R}\implies$ connected

Show that an interval in $\mathbb{R}$ is a connected set in $\mathbb{R}$. Edit: A different proof attempt. Let $A$ be an interval in $\mathbb{R}$ and let $a=\sup(A)$ and $b=\inf(A)$. Suppose ...
2
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3answers
350 views

Why is boundedness defined so differently in a topological vector space and in a metric space?

From Wikipedia A subset S of a metric space (M, d) is bounded if it is contained in a ball of finite radius Also from Wikipedia a set in a topological vector space is called bounded or von ...
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2answers
86 views

Geometric explanation of the product metric

Can someone describe to me the geometric intuition behind using a mapping $$ ((x_1,y_1),(x_2,y_2)) \mapsto \frac{d_1(x_1,y_1)}{1+d_1(x_1,y_1)} + \frac{1}{2} \frac{d_2(x_2,y_2)}{1+d_2(x_2,y_2)} $$ to ...
1
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2answers
81 views

Compactly supported function?

I'm doing some self-study and I ran into a situation as follows. Suppose $(X,d)$ is a metric space and $F\subset X$ is compact. For some $\varepsilon>0$ let $V=\{x:d(x,F)<\varepsilon\}$. Is the ...
2
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1answer
64 views

$f(x)=x^n,n\in\mathbb{N}$, exists unique $a$ s.t. $a^n=b$

Let $(\mathbb{R},d_2)$ be a metric space, and let $f:\mathbb{R}\rightarrow\mathbb{R}$ be given by $f(x)=x^n$ for $n\in\mathbb{N}$. If $b\in\mathbb{R_+}$, show that there exists a unique ...
2
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1answer
103 views

Definition of equicontinuity of a mapping between metric spaces

From Wiki Let $X$ and $Y$ be two metric spaces, and $F$ a family of functions from $X$ to $Y$. The family $F$ is equicontinuous at a point $x_0 ∈ X$ if for every $ε > 0$, there exists a $δ ...
2
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0answers
126 views

What's the relationship between the riemannian metric and Jacobi field?

I encounter to the question in reading the following Excise: Let $(M,g)$ be a $m$-dimensional Riemannian manifold, and $(r,\theta^1,\theta^2,\ldots,\theta^{m-1})$ be the (geodesic) polar ...
2
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2answers
120 views

Showing that a sequence converges (in metric space)

In $(\ell ^\infty,{\Vert .\Vert_\infty)}$, how would I show that $x_n=\left(\frac{n+1}{n},\frac{n+2}{2n},\frac{n+3}{3n}, ...\right)$ converges and how would I find the limit? I tried using the fact ...
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2answers
2k views

Give an example of a topological space which is not a metric space, i.e. whose topology is not associated with any metric.

Give an example of a topological space which is not a metric space, i.e. whose topology is not associated with any metric.
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1answer
764 views

Show $f(x)=x\ln x$ is not uniformly continuous

Show $f:(0,\infty)\rightarrow \mathbb{R}, f(x)=x\ln x$ on $(0,\infty)$ is not uniformly continuous. I think that the general way to prove that something is not continuous in a metric space is to ...
2
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2answers
68 views

Show $(A^o)^c=\overline{A^c}$

Show $(A^o)^c=\overline{A^c}$. ($\rightarrow$) $(A^o)^c\subseteq\overline{A^c}$ I want to show that $(A^o)^c$ is closed and that $A^c\subseteq (A^o)^c$. Then ($\rightarrow$) follows. Since ...
2
votes
2answers
112 views

$A$ is open $\iff$ $A\cap\partial A=\emptyset$

Show $A$ is open $\iff$ $A\cap\partial A=\emptyset$. Attempt: ($\rightarrow)$ $A$ open $\implies A\cap\partial A= \emptyset$. $x\in A$ open ...
0
votes
2answers
75 views

Proving $A^o=A\setminus \partial A$

Show $A^o=A\setminus\partial A$. Attempt: I need to show inclusion on both sides, so: (a) $A^o\subseteq A\setminus\partial A$ (b) $A\setminus\partial A \subseteq A^o$ Attempt at (a): If ...
3
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1answer
153 views

How do I show that this metric space is not convex?

Denote $X$, the space of all sequences $\in$ $\mathbb R$. I have a metric $$d(x,y):=\sum_{n=1}^\infty 2^{-n}\frac{| x_n-y_n|}{1+| x_n-y_n|}$$ and $(X,d)$ is a metric space. How would I show that the ...
1
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1answer
105 views

Showing boundedness of metric space

Denote $X$, the space of all sequences $\in$ $\mathbb R$. I have a metric $$d(x,y):=\sum_{n=1}^\infty 2^{-n}\frac{| x_n-y_n|}{1+| x_n-y_n|}$$ If $(X,d)$ is a metric space and if $A$ is a subset of ...
7
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1answer
5k views

Continuous function on a compact metric space is uniformly continuous

I am struggling with this question: Prove or give a counterexample: If $f$ is a continuous function on a compact subset $Y$ of a metric space $X$, then $f$ is uniformly continuous on $Y$. ...
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2answers
64 views

Is the set $E$ of sequences containing only entries $0$ and $1$ in $(m,\left \| \cdot \right \|_\infty)$ complete?

I can't really wrap my head around $E$, or a Cauchy sequence in $E$. I need to take a Cauchy sequence in $E$ and show it's Cauchy in $(m,\left \| \cdot \right \|_\infty)$? I think I can show $(m,\left ...
1
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0answers
94 views

Distinct metrics on a manifold

I'm trying to understand basic differential geometry (my background is in mathematical logic), and I'm having a bit of difficulty with a basic point. Frequently we want to consider the set of metrics ...
2
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3answers
668 views

Let S be a subspace of topological space X. Show that the closure of S, the set of contact points, is indeed closed.

Let $S$ be a subspace of topological space $X$. Show that the closure of $S$, the set of contact points, is indeed closed. I need to prove that the closure is closed but I don't know how to ...
3
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1answer
2k views

totally bounded, complete $\implies$ compact

Show that a totally bounded complete metric space $X$ is compact. I can use the fact that sequentially compact $\Leftrightarrow$ compact. Attempt: Complete $\implies$ every Cauchy sequence ...
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2answers
1k views

$X$ compact metric space, $f:X\rightarrow\mathbb{R}$ continuous attains max/min

Let $X$ be a compact metric space, show that a continuous function $f:X\rightarrow\mathbb{R}$ attains a maximum and a minimum value on $X$. Attempt: So the important thing is that I have ...
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3answers
147 views

Proving this function is a metric

This is a space $S$ that consists of the set of all sequences of real numbers and $x=(x_1,x_2,x_3,...), y=(y_1,y_2,y_3,...)$ etc. and the metric $d$ is defined as $$d(x,y)=\sum_{i=1}^\infty ...