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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Show that if $(X,d)$ is compact then, every open covering of $X$ has a Lebesgue number.

Let $(U_i)_{i \in I}$ be an open cover of a metric space $(X,d)$, a number $\epsilon >0$ is called a Lebesgue number of $(U_i)_{i \in I}$ if for all $x \in X$ exist $j \in I$ such that ...
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1answer
23 views

Prove that $G(f)$ is homeomorphic to $X$.

Let $X,d$ be a metric space .Let $f:X\to \mathbb R$ be a continuous function.Define $G(f)=\{(x,f(x)):x\in X\}$. Prove that $G(f)$ is homeomorphic to $X$. My try: Since $f$ is continuous then $G(f)$ ...
0
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1answer
36 views

Is this condition on continuity extraneous or troublesome?

I was trying to motivate the use of open sets for defining continuity (as in topology or metric spaces). I came to formulate the following definition of continuity for a function $f: X \rightarrow ...
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19 views

Metric spaces and compactness [on hold]

Let $X$ be a metric space. If for all compact $K$, the set $K\cap F $ is closed, then $F$ is closed.
3
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4answers
261 views

Little confusion about connectedness

Consider $X=\{(x,\sin(1/x)):0<x<1\}$. Then clearly $X$ is connected , as it is a continuous image of the connected set $(0,1)$. So, $\overline X$ is also connected , as closure of connected set ...
0
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2answers
27 views

Find the limit points and exterior points of the following

Let $X=\mathbb R$, with the usual metric on $\mathbb R$ and $A=((0,1)\cap \mathbb Q)\cup$ {$2,3$}. Find the limit points of $A$, exterior points of $A$, $A^o$, $\overline A$ and $\partial A$. Can ...
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2answers
60 views

Is $[-1,1]$ complete under the Euclidean metric? [on hold]

Is it true that the interval $[-1,1]$ is complete under the Euclidean metric?
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31 views

Show that $x_n \to x$ in $(X,d_f)$ iff $f(x_n) \to f(x)$

Let $(x_n)$ be a sequence in $X$ and $x \in X$. $(X,d)$ is a metric space and $f: X \times X\to $ is a bijection. Show that $x_n \to x$ in $(X,d_f)$ iff $f(x_n) \to f(x)$. ...
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2answers
31 views

Show that $d_f$ is a metric on $X$ [on hold]

Let $(X,d)$ be a metric space, and let $f: X \to X$ be a bijection. Define $$d_f: X \times X \to \mathbb R $$ as $d_f(x,y)=d(f(x),f(y))$ $\forall x,y \in X$ Show that $d_f$ is ...
0
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1answer
38 views

What exactly is the distance of two elements in $C[0,1]$?

If $C[0,1]$ — the set of all continuous functions from $[0,1] \rightarrow \mathbb R$ — is equipped with the metric $||\cdot||_1$ (1-Norm), then what is the distance between ...
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2answers
50 views

Bounded complete metric space is compact?

This question may seem trivial, but in topology we were taught that in a complete metric space, a subset of that space was compact if and only if it is closed and bounded. Moreover, we are told that ...
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0answers
36 views

Is $ \text{Int} \overline{B(a;r)} = B(a;r)$ for a metric space $(X,d)$?

I think this is true in general. To give a brief outline of a proof: Let $ \text{Int} \overline{B(a;r)} = U $, I claim that if $a \in U \implies a \notin Fr(B(a;r))$ so $a \in \text{Int}B(a;r)$ ...
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0answers
21 views

Making a metric out of distance measure

I'm working with a pseudo-distance measure that is not a metric since it does not hold the triangle inequality. It is called Dynamic Time Warping. The problem is - I need to perform some projections, ...
5
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1answer
69 views

If every real valued continuous function on $X$ is uniformly continuous , then is every continuous function to any metric space uniformly continuous?

Let $X$ be a metric space such that every continuous function $f:X \to \mathbb R$ is uniformly continuous ( here $\mathbb R$ is equipped with the standard euclidean metric ) , then is it true that for ...
3
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2answers
57 views

If $\overline A\cap B = A\cap \overline B = \varnothing$, $A\cup B$ is disconnected.

I'm trying to show that if $\overline A\cap B = A\cap \overline B = \varnothing$, $A\cup B$ is disconnected. First of all, I think I have to assume that $A$ and $B$ are nonempty, or else the statement ...
0
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1answer
23 views

Is this a metric on $\mathbf P\mathcal H$?

Let $\mathcal H$ be a real or complex Hilbert space with inner product $\langle\cdot,\cdot\rangle$. On the projective space $\mathbf P\mathcal H = \left(\mathcal H\setminus\{0\}\right)\big/{\sim}$ ...
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1answer
31 views

Normed space of bounded functions $f:\mathbb{N}\to\mathbb{N}$

Let $X = \{f:\mathbb{N}\to\mathbb{N}: \exists M\in\mathbb{N} \forall n\in\mathbb{N} f(n) \leq M\}$. Define a norm on $X$ by defining for $f\in X$: $$||f|| = \sum_{n=1}^\infty \frac{f(n)}{2^n}.$$ Is ...
2
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0answers
38 views

Prove that a finite union of closed sets is also closed (using limit points)

Let $F_i$ be a family of closed sets, then we know that $\bigcup_{i=1}^nF_i$ is closed. Proving that statement is equivalent to proving: If $p$ is a limit point of $\bigcup_{i=1}^nF_i$ then ...
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1answer
33 views

If $f:X \to [0,1]$ is an onto continuous closed map and $\{f^{-1} (y)\}$ is compact for every $y \in [0,1]$, then is X necessarily compact?

If $f:X \to [0,1]$ is an onto continuous closed map and $\{f^{-1} (y)\}$ is compact for every $y \in [0,1]$, then is X necessarily compact? Now continuous image of a compact set is compact. Again ...
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0answers
31 views

Lower semicontinuity on a metric space

I'm trying to prove something about lower semicontinuity for a map on a metric space $(X,d)$. I will try to write here my idea of the proof, hope someone can approve or contest it. Def. Let $(X,d)$ ...
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1answer
58 views

Dense-in-itself open sets in a subspace of the real line

Given an uncountable set $X\subset [0,1]$ it is easy to write $X$ as a disjoint union of a perfect set $P$ (perfect in the subspace $X$) and an at most countable set $C$: just take $P$ as the set of ...
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2answers
56 views

If $f:X \to [0,1]$ be an onto continuous map and $\{f^{-1} (y)\}$ is compact then Is $X$ compact?

If $f:X \to [0,1]$ is an onto continuous map and $\{f^{-1} (y)\}$ is compact for every $y \in [0,1]$, then is X necessarily compact? Now continuous image of a compact set is compact. Again $X$ is ...
1
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1answer
37 views

How to prove a function is continuous on a compact set?

I´m struggleing with this problem: I know by theorems that inf(d(a,b)) exists if the real value function d is continuous on the set AxB. But how can I prove that d is continuous?
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33 views

Topologically equivalent metrics, using different definitions.

I´ve been dealing with topologically equivalent metrics for a while, using the usual definition, that $d$ and $d'$ are topologically equivalent iff they have the same open sets. However, there is ...
3
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1answer
35 views

Let $f: (X, d) \to (Y, d')$. Prove that the following are equivalent:

Let $f: (X, d) \to (Y, d')$. Prove that the following are equivalent: a) $f$ is uniformly continuous in $X$. b)For every pair of sequences $(x_n), (y_n) \subseteq X$ such that $ d(x_n, ...
3
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1answer
35 views

Show that $A=\bigcap G_{A}$

Given a metric space $(X,d)$ and $A\subset X$, let $G_{A}$ be the set which consists of all the open sets that contain $A$. Show that $A=\bigcap_{B \in G_{A}}B$ It is obvious that $A \subset ...
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Is $d_1(x,y):= x^2-y^2$ a metrics on $\mathbb{R}$? [on hold]

$$d_1(x,y) = x^2-y^2 \quad \forall x,y \in \mathbb{R}$$ Is $d_1$ a metric on $\mathbb{R}$?
0
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1answer
19 views

show that $d(a,b)\leq r-s-t \Rightarrow K(c,t) \subseteq K(a,r)$ in a metric space, assuming that $r,s,t>0$ and $c \in K(b,s)$.

Alright, so in a metric space, $M$, with $r,s,t>0$, $a,b,c \in M$ and $c \in K(b,s)$ I have to show, that: $d(a,b)\leq r-s-t \Rightarrow K(c,t) \subseteq K(a,r)$ I really have no idea where to ...
2
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1answer
25 views

Topological field - Proving continuity of inversion

Given a field $F$ and an absolute value $|\ |$ on $F$, define the distance $d(x,y)$ between two elements $x,y\in F$ by $$ d(x,y) = |x - y|. $$ I just worked through the proofs that $d$ defines a ...
0
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1answer
27 views

Dissimilarity and the triangle inequality

Suppose the degree of dissimilarity between $a$ and $b$ is their proportion of properties not in common (the number of $a$ alone and of $b$ alone divided by the total number of $a$ and $b$). I want to ...
1
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1answer
20 views

Discrete metric, countable basis?

Give an example of a metric space which does not have a countable basis. I was thinking of some uncountable set, with a metric which results in an uncountable number of open subsets. Which ...
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3answers
33 views

Inequality leading to Holder's: $t^\theta \leq \theta t + 1 - \theta$ with $0<\theta<1$ and $t\geq 0$

In the process of proving Holder's Inequality for $l^p$ spaces, as per my instructions it begins by first asking us to prove the following inequality as a first step: If $0<\theta<1$ and ...
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2answers
44 views

Show that a discrete metric space is finite. [on hold]

Problem: Is every discrete metric space finite? I know that every subset of a discrete metric space is both open and closed but I'm stuck otherwise. Please help!
2
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1answer
23 views

Is AC necessary to show that in metric spaces $x\in\operatorname{closure}(A)$ implies $\exists\{a_n\}_{n=1}^\infty\subseteq A$ s.t. $\lim a_n=x$?

Let $(X,d)$ be a metric space. Let $x\in\operatorname{closure}(A)$, where $A\subseteq X$. Then for each $n\in\mathbb{N},\exists x_n\in B_{\frac{1}{n}}(x)\cap A$, where $B_\varepsilon(x)$ is the open ...
4
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1answer
57 views

Good function's

I'm trying to solve the following question: Let $(X,d)$ be a metric space. We call a continuous function $f:X\to \mathbb R$ "good function" if for every continuous function $g:X\to \mathbb R$, ...
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13 views

Show that the mobius transformation, $M:H \rightarrow H$ is a homeomorphism

Where $M$ is defined: $M(z) = \frac{ez+f}{gz+h}$ And, $H = \{z= x+iy \in \mathbb{C} \space| x,y \in \mathbb{R}, y > 0 \}$ is the upper half plane, with the induced topolgy such that the ...
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2answers
41 views

Q: Nowhere dense sets.

Given $X$ a metric space, $A\subset X$ a nowhere dense set. Show that every open ball $B$ contains another open ball $B_1 \subset B$ such that $B_1 \cap A = \emptyset$. EDIT: I modify my proof ...
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51 views

Compact subset of an open set in the complex plane

I would just like to ask a question. Suppose $K$ is a compact subset of an open set $V$ in the complex plane, how would you prove that there exists an $r > 0$ such that the union $E$ of the closed ...
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1answer
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question in metric space completion properties

Dear all I was given this question in topology which I would really appreciate help with: I am asked to prove that for every metric space we have that the space itself is totally bounded if and only ...
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2answers
33 views

Show that a countable dense subset $D \subset X$ is not a $G_{\delta}$

Given $X$ a complete metric space with no isolated points and $D \subset X$ a countable dense subspace, show that $D$ is not a $G_{\delta}$. I am quite lost in trying to use the hypothesis of the ...
3
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2answers
43 views

Euclidean distance on R and Q

I have to answer the following three questions. Given $\epsilon > 0$, a metric space $(V,d)$ and a element $a \in V$ i) Show that for all $x \in B(a;\epsilon)$ there exists a $n \in \mathrm{N}$ ...
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1answer
31 views

Is there a name for this property of a subset of a metric space?

Our 'intro to Analysis' professor intruduced compactness of subsets of metric spaces by first introding what he calls 'rijcompact' (sequential compactness), and going on to prove that it is equivalent ...
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0answers
25 views

Notation for Christoffel symbols used by Gödel in “An example of a new type of cosmological solution of Einstein field equations of gravitation”

I have difficult to understand the meaning of the notation used by Gödel in the article cited in the title of this post. You can find it here: http://www.lygeros.org/10552b.pdf In the second page ...
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1answer
178 views

In a metric space, is every convergent sequence bounded?

In $\mathbb{R}$ and $\mathbb{R}^p$, this is true, but is it true in every metric space? I suppose not, but what other condition would I have to put on the metric space in order for it to have this ...
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2answers
20 views

Question regarding metric spaces and open balls

I have just started with metric spaces and am reading Karl Stromberg's Introduction to Real Analysis. In one of the examples, the following is stated: If $X=\mathbb N\subset \mathbb R$ with the ...
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1answer
49 views

Show that $\mathbb{R}^n$ cannot be written as a countable union of proper subspaces

Show that $\mathbb{R}^n$ cannot be written as a countable union of proper subspaces Ok so I know I have to use Baire's Cathegory Theorem here. And I've done the following, lets suppose on the ...
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1answer
48 views

The metric space and the balls.

Let $M$ be a metric space with a distance function $d$, and let $a,b,c\in M$ be given. Let $r,s,t>0$ and assume that $c\in K(b,s):=\lbrace x\in M\mid d(b,x)<s\rbrace$. a) Show that if ...
0
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1answer
25 views

A simple version of the Picard-Lindelöf Theorem

I wish to ask a particular question about following the proof in this theorem, and thought the best place to come might be here. It is as follows: First, we have a differential equation that ...
2
votes
2answers
60 views

Existence of a metric space where each open ball is closed and has a limit point

Show that there exists a metric space in which every open ball is closed and contains a limit point. I think that the space $\{\frac{1}{n}\mid n\in\mathbb{N},n>0\}\cup \{0\}$ with the standard ...
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1answer
32 views

Show that $\exists a\in A; b\in B$ such that $d(a,b)=d(A,B)$

Let $A,B$ be two compact subsets of $X$ where $(X,d)$ is a metric space. 1.Show that $\exists a\in A; b\in B$ such that $d(a,b)=d(A,B)$ where $d(A,B)=\sup\{d(a,b):a \in A;b\in B\}$ 2.Show that ...