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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Proving homeomorphisms

I came across this practice question, which seems rather simple - but I am wondering if I am not understanding something completely. If I were to define an explicit homeomorphism to demonstrate that ...
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Existence of a continuous function which does not achieve a maximum.

Suppose $X$ is a non-compact metric space. Show that there exists a continuous function $f: X \rightarrow \mathbb{R}$ such that $f$ does not achieve a maximum. I proved this assertion as follows: ...
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How to prove the triangle inequality for this distance?

I'm studying a proof in 'An Introduction to Metric Spaces and Fixed Point Theory' (M. Khamsi, W. Kirk) that shows the equivalence of injectiveness and hyperconvexity for metric spaces. I stumbled over ...
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Continuity of metric function

Let $X$ is a vector space and $d$ is a metric function on $X$ and $\|\cdot\|$ is a norm on $X$ and $\langle\cdot,\cdot\rangle$ is an inner product function on $X$ It is to easy to prove $\|\cdot\|$ ...
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Show restriction map is a contraction/lipschitz mapping

For $C[a,b]$ (set of all continuous real valued functions), define $d(f,g) = \int^{b}_{a}|f(x)-g(x)|dx$ If $[c,d]$ is a subinterval of $[a,b]$ and the mapping $r:C[a,b] \rightarrow C[c,d]$ ...
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Show that a set is not open

Suppose $U_1$ and $U_2$ are both nonempty subsets of $\mathbb R$ such that $U_1 \cap U_2 =\emptyset $ and $U_1\cup U_2 = \mathbb R.$ Consider points $p \in U_1\ \text{and}\ q \in U_2.$ Without loss ...
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44 views

Prove that this is a metric space

The following is a question from a previous assignment that I was unable to complete. Any assistance on how to complete this would be appreciated. Let $\varrho: X\times X\to \Bbb R^+$ be a metric on ...
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Spaces in which “$A \cap K$ is closed for all compact $K$” implies “$A$ is closed.”

Let $X$ denote a topological space. For any $A \subseteq X$, consider two possible conditions on $A$. $A$ is closed $A \cap K$ is closed, for all compact $K \subseteq X$. If $X$ is Hausdorff, then ...
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A subset of a metric space is closed iff its intersection with every compact subset is closed

I want to show that a subset of a metric space $X$ is closed iff its intersection with every compact subset of $X$ is closed
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Prove that this infinite sum involving metrics is also a metric

The following is a question from a previous assignment that I was unable to complete. Any assistance on how to complete this would be appreciated. Let $\varrho_i: X\times X\to \Bbb R^+$ with ...
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Convergence in Fréchet spaces and the topology

actually i have two questions concerning Fréchet spaces (i am not familiar with these spaces so i need some help). I have a vector space $V$ with a distance $$d(x,y) = \sum_{n\in \mathbb{N}} ...
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Show that $d(x,y)=\frac{|x-y|}{1+|x-y|}$ is a metric on $\mathbb{R}$ (triangle inequality) [duplicate]

Prove that $d(x,y)=\frac{|x-y|}{1+|x-y|}$ is a metric on $\mathbb{R}$. Definition. A function $d:E \times E \mapsto [0, \infty)$ is called a metric iff whenever $x,y,z \in E$, $d(x,y) = 0$ if ...
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43 views

Compactification of a straight line

Like in the case of mapping a infinite-plane to a sphere (Riemann Sphere), I can understand, that I can map the infinite line ($-\infty,\infty$) to a circle. Secondly, I can also map a finite line ...
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Question on p-adic norms and metric spaces

I was given this assignment question: I am given two distinct primes $ p \neq q $ and I am asked to produce first of all a set closed with respect to p-adic norm but not closed under q-adic norm. The ...
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$d_1(x,y)=|x-y|$ & $d_2(x,y)=|\frac1x-\frac1y|$ inducing same topology on $[1,\infty)$?

My textbook says the following: Let $X=[1,\infty)$ and define $d_1(x,y)=|x-y|$ and $d_2(x,y)=|\frac1x-\frac1y|$ . Then $d_1$ and $d_2$ are inducing the same topology, $(X,d_1)$ is complete but ...
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What is the interior of a single point in a metric space?

Let $(X,d)$ be a metric space. We know that if $x \in X$ , then $Cl(\{x\})=\{x\}$, which implies that $\{x\}$ is closed. However if that's the case, what would the interior of $\{x\}$ be? I was ...
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Checking my understanding of the Interior of these intervals

Let $[a,b]$ be any finite closed interval. (i) $\text{Int}_{[a,b]}(a,b]$ Am I correct to say that the interior of this set is $[a,b]$? Since the interior of a set are all the points in the set in ...
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Metric Fixed Point Theory

I am learning Metric Fixed Point Theory by Mohammed A Khamsi and William A Kirk. I need help in understanding a step in the proof of the following theorem(Chapter 3, Theorem 3.2, Page No. 43): ...
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$\{\infty\}$ open in $\mathbb N\cup\{\infty\}$ with $d(a,b)=|\arctan a-\arctan b|$?

Let $X=\mathbb N\cup\{+\infty\}$. I want to find two metrices inducing different topologies. Let $d_1$ be the discrete metric then all subsets of $X$ are open. (in particular $\{+\infty\}$) But now ...
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Characterization of the circle within metric spaces

There are various characterizations of the circle. To be precise, there is not the circle. There are several categories which contain an object we refer to as "the circle". In $\mathsf{Top}$ the ...
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Showing Lipschitz continuity for a particular distance functiom.

My friend and I have been working on trying to prove this inequality for awhile, however, I think there is some trick we are just not seeing. Suppose $F$ is a closed set in $\mathbb{R}$, whose ...
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Showing a function $f:X \rightarrow Y$ is continuous

I am working through some practice questions, and I am not sure if I am on the right track with this one: Let $X = \cup_{n≥1}A_n$, be a topological space and assume that a map f : X → Y is such ...
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Give a example of a sequence of continuous functions which do not form a Cauchy sequence

As an example that not every Cauchy sequence in $(M,d)$ is converging in $M$ the following examples are given: Consider $(\mathbb{Q},d_{\text{eucl}})$ and a sequence $q_n \in \mathbb{Q}\to ...
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Metric spaces are completely normal

Given a metric space $(X, k)$ with $Y, Z\subset X$ and $\operatorname{cl}(Y)\cap Z = \emptyset$, $\operatorname{cl}(Z)\cap Y = \emptyset$, prove that there are open sets $M, N$ such that $Y\subset ...
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If $E = \bigcup_{n = 1} ^ \infty \left(\frac{1}{n+1}, \frac{1}{n}\right) $. Show $\bar E = [0,1]$ is the closure of $E$

If $E = \bigcup_{n = 1} ^ \infty \left(\frac{1}{n+1}, \frac{1}{n}\right) $. Show $\bar E = [0,1]$ is the closure of $E$ Attempt: Since $ (\frac{1}{n+1}, \frac{1}{n}) $ is a subset of [0,1], so $E ...
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What are some illustrative examples that demonstrate how $\succ$ can differ in behavior from $>$ and/or $\geq$?

I really, really want to understand the generalization of metric spaces known as continuity spaces. Unfortunately, I always get tripped up right at the beginning. The problem is that I have little or ...
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Open Sets in $\mathbb{R}$

I was wondering what the general form of an open set is in the real numbers. Is it just an interval of the form $(a,b)$; $a,b \in \mathbb{R}$.
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Sequence characterization of bounded sets

If $M$ is an arbitrary metric space, the following holds: $A\subseteq M$ is totally bounded $\Leftrightarrow$ Each sequence in $A$ contains a Cauchy subsequence. Additionally, for ...
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If two metrics are equivalent and one is totally bounded, is the other totally bounded?

I want to know if the following proof is correct... If $(X,d)$ is separable then, if $S$ is an open cover of $X$, I can pick a numerable number of open sets in $S$, such that $X$ is included in their ...
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The projection map $\pi_j : \Bbb R^n \to \Bbb R$ given by $\pi_j(x) = x_j$ is open but not closed.

The projection map $\pi_j : \Bbb R^n \to \Bbb R$ given by $\pi_j(x) = x_j$ is open but not closed. I have found an example for the map not to be closed. But unable to prove that it is open. Please ...
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How do you prove that a metric space $X$ is separable if and only if $X$ has a countable subset $Y$ with property below?

A metric space $X$ is separable if and only if $X$ has a countable subset $Y$ with property: for $\epsilon > 0$ and every $x \in X$, there is a $y \in Y$ such that $d(x, y) < \epsilon$.
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Does there exist non-compact metric space $X$ such that , any continuous function from $X$ to any Hausdorff space is a closed map ?

I know that there is a topological space $X$ which is not compact but such that , for any Hausdorff topological space $Y$ , any continuous function $f:X \to Y$ carries closed sets to closed sets . I ...
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If $E$ is closed and bounded in a metric space $X$ and $f: E → R$ is continous on $E$ , prove that $f$ is bounded on $E$.

If $E$ is closed and bounded in a metric space $X$ and $f: E → R$ is continous on $E$ , prove that $f$ is bounded on $E$. and suppose that $X$ satisfy the Bolzano Weierstrass Property attempt: ...
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Doubt related to the extended real line and distance/metric

I am studying real analysis and I am being introduced to functions that take values on the extended real line, I have a fundamental doubt about this so I'll give an example to illustrate my confusion: ...
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Maps from $D^n$ to $D^n$ with a single inverse set are open.

Let $D^n$ denote the closed unit ball in $\Bbb R^n$. In multiple sources proving Brown's generalized Schoenflies theorem (including a version in the original paper), the following consequence of ...
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The square $S := [- R, R] \times [-R, R]$ is a compact subset of $\Bbb R^2$.

The square $S := [- R, R] \times [-R, R]$ is a compact subset of $\Bbb R^2$. An intuitive approach: Let $S$ be not compact then there is an open cover of which there is no finite sub cover of $S$.Now ...
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2answers
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If $g: Y \to Z$ is a continuous injection, then a map $f : X \to Y$ is open if $g\circ f$ is open.

If $g: Y \to Z$ is a continuous injection, then a map $f : X \to Y$ is open if $g\circ f$ is open. To show the map $f : X \to Y$ is open, we first take any open subset $U$ from $X$ and then show that ...
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Is $\overline{\mathbb{R}}^+$ a compact Polish space

if $X$ is defined by $$X= [0,+\infty)\cup\{+\infty\}$$ is endowed with the metric $$d_X(x,y) = |\arctan(x) - \arctan(y)|$$ Is it true that the metric space $(X,d_X)$ meets the following properties? ...
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Sequence has a convergent subsequence in R^n

Suppose A is a closed and bounded subset of R^n. Let {ak} be a sequence in A. Thus, the elements of {ak} are: (a11,a12,...,a1n), (a21,a22,...,a2n), ... ... (ak1,ak2,...,akn), ... We are not sure if ...
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Alternate definition limit

The definition of the limit of a sequence is: $L=\lim\limits_{n\to\infty}f(n)\Leftrightarrow\forall\epsilon>0:\exists N\in\mathbb{N}:\forall n\in\mathbb{N}:\left(n>N\Rightarrow ...
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Name of the metric: $d(f,g)=\max \limits_{a\leq x \leq b} |f(x)-g(x)|$

What is the name of the metric: $$d(f,g)=\max \limits_{a\leq x \leq b} |f(x)-g(x)|$$ Where $f,g\in X$ where $X$ is the space of all continuous functions. I can't find any documentation on this ...
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Every compact metric space is complete

I need to prove that every compact metric space is complete. I think I need to use the following two facts: A set $K$ is compact if and only if every collection $\mathcal{F}$ of closed subsets with ...
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1answer
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An example of a dense and co-dense set in a metric space with countable derived set

Let $(X,d)$ be a metric space and $A\subset{X}$ such that $A$ and $A^c$ are both dense in $X$. Show that it is not necessary that $A^\prime$ be uncountable. And prove $(A^\prime)^\prime=A^\prime$. ...
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Compact subset of an open set

Let $X$ be complete separable metric space and $A\subset X$ is open. Does it mean that there is a compact subset of $A$? My solution is the following: since $A$ is open there is $B(x,r)\subset A$, ...
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1answer
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A complete subset of a metric space is closed?

Supposing $A$ is a subset of a metric space $S$, it is simple enough to show that if $S$ is complete and $A$ is closed, that $A$ is complete. However, without being given that $S$ is complete, what ...
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2answers
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If $A$ is connected , $B$ is open and closed and $A \cap B$ is non empty then $A \subset B$.

Let $(X,d)$ be a metric space and $A,B \subset X$. If $A$ is connected , $B$ is open and closed and $A \cap B$ is non empty then $A \subset B$. I tried it with proving a contradiction if we first ...
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Show that a point $ a \in X$ is not a cluster point of $X$ if $a$ is isolated.

A point $a$ in a metric space $X$ is said to be isolated if and only if $r> 0$ so small that $B_r(a)$ = {$a$} Show that a point $ a \in X$ is not a cluster point of $X$ if $a$ is isolated. proof: ...
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1answer
24 views

Completion of a sequence space

Let $F$ be a field with some absolute value $|\cdot|$. Consider the space $X$ of sequences $\mathbf{a} = (a_1, a_2, a_3, \cdots)$ for which $a_i \in F$ for all $i\in\mathbb{N}$ and at most finitely ...
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Question about contraction mapping (an easy example) [closed]

Is following map a contraction on the real line? $$ f(x)=x+\frac{1}{1+e^{x}}. $$ If yes then what is the contraction coefficient?
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2answers
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Prove that a compact metric space is complete.

I'm reading Intro to Topology by Mendelson. I'm in the section titled "Compact Metric Spaces". The problem is in the title. My attempt at the proof is as follows: Let $\{a_n\}_{n=1}^\infty$ be a ...