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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60 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 ...
5
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1answer
70 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 ...
2
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1answer
24 views

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
25 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 $G(f)$ ...
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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
265 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 ...
2
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0answers
108 views

$A \subset \Bbb R$ such that $A$, $clA$, $int(A)$, $cl(int(A))$, $int(clA)$ are pairwise distinct

Do there exist subsets with internal closures $A$ of $\mathbb R$ such that $A$ , $\bar A$ , $A^\circ$ , $(\bar A)^\circ$ , $\overline{A^\circ}$ are pairwise distinct? I found an example from a book ...
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2answers
28 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 ...
2
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2answers
80 views

Prove that the set $X+Y$ is closed

Consider two sets: $$X=\bigl\{(x,y)\in \mathbb R^{2}:xy=1\bigr\}$$ $$Y=\bigl\{(x,y)\in \mathbb R^{2}:|x|\le 1,|y|\le 1\bigr\}.$$ We find that , both the sets are closed. But, is the set $X+Y$ ...
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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
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 ...
2
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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
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
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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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, ...
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 ...
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1answer
20 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 ...
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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
71 views

Give a counter example to show that given two metrics are NOT equivalent.

Finding difficult to find a counterexample show that two metrics are not equivalent. Set: $C[0,1] $ of all continuous functions on the interval $[0,1]$. Metric 1: $d(x,y) = \max\limits_{t \in [0,1]} ...
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0answers
33 views

Contracting subsets

Let $X$ be a (locally finite) metric graph (all of whose edges are length 1). A subset $A \subset X$ is contracting if there exists a constant $C \geq 0$ such that the nearest point projection on $A$ ...
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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 ...
1
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1answer
35 views

Metric spaces and fix point [closed]

I saw this problem in my course of Intr. to the topology: Let $(X,d)$ be a compact metric space and $$f :(X,d) \rightarrow (X,d)$$ a continuous function such that: $\quad d(f(x);f(y)) < ...
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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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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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2answers
57 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 ...
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0answers
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 ...
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1answer
59 views

$A$ and $B$ compact in a Hausdorff space implies $A\cap B$ is compact [closed]

Prove that if $A$ and $B$ are compact subset of a Hausdorff space $X$, then $A$$\cap$$B$ is compact.
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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2answers
35 views

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 $\mathbb{R}$?
3
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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 ...
6
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1answer
258 views

Basic questions about $\mathbb{Z}^{\mathbb{N}}$ with the product topology

can someone please let me know if the following is correct: 1) Let $\mathbb{Z}$ be the integers endowed with the discrete topology and $\mathbb{N}$ the natural numbers. Is $\mathbb{Z}^{\mathbb{N}}$ a ...
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 ...
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2answers
599 views

Extension of a Uniformly Continuous Function between Metric Spaces

Let $(X,d)$ and $(Y,d')$ be metric spaces with $(Y,d')$ complete. Let $A\subseteq X$. I need to show that if $f:A\to Y$ is uniformly continuous, then $f$ can be uniquely extended to $\bar{A}$ ...
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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 ...
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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2answers
44 views

Show that a discrete metric space is finite. [closed]

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!
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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 ...
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 ...
3
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2answers
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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2answers
1k views

Triangle Inequality for supremum metric

Edited Heavily Here all functions are from $[0,1]$ to $\mathbb{R}$ and are bounded. Prove the following Triangle inequality in following case: ...
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0answers
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
75 views

Prove that $f^{-1} (F)$ is closed

A set $F \subset \mathbb R$ is closed if for any convergent sequence $\{x_n\}$ in F converges, we have $\lim_{n \to \infty} x_n=x \in F $. How to Prove that if $f :\mathbb R \to \mathbb R$ is ...
0
votes
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 ...
0
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1answer
20 views

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 ...
2
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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 ...