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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$X$ is A-space iff the frontier of any closed set in $X$ is compact.

Hi everyone I have troubles with the following proposition: Definition: We say a metric space $(X,d)$ is an A-space iff every Hausdorff image of $X$ under a closed continuous map is metrizable. ...
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392 views

Help sketching 'Jungle River Metric' in $\mathbb{R}^2$

I just need to clarify that i've sketched these open balls correctly, the metric is given by: $$d(x,y) = \begin{cases} |x_2-y_2|, & \text{if $x_1 = y_1$;} \\ |x_2| + |y_2| + |x_1-y_1|, & ...
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Example of metric continuous with respect to another metric but generating different topology

Take, say, the standard 2-sphere $S^2$. Equip it with some metric $d$; this metric will generate a topology that may or may not coincide with the standard Euclidean topology. In the case it does, ...
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Prove that there exists $y_0\in C$ such that $d(x,y)=\inf_{y\in C} d(x,y)$, i.e. $y_0$ is a closest point to $x$ in $C$.

If $C$ is a closed subset of $R^n$ and $x\in R^n$, prove that there exists $y_0\in C$ such that $d(x,y)=\inf_{y\in C} d(x,y)$, i.e. $y_0$ is a closest point to $x$ in $C$. Here's what I got but ...
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63 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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2answers
32 views

Definition of open ball in discrete metric space

I would like some help clarifying the definition of open balls in the discrete metric space. The definition I am provided is: Open balls in the discrete metric space $M = (X,d_0) $ are given by ...
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20 views

Prove compact metric spaces $X$ and $Y$ are isomorphic given these conditions [duplicate]

Let $X$ and $Y$ be compact metric spaces and for each finite subset $A$ of $X$ there is a finite subset $B$ of $Y$ such that A is isometric to B and for each finite subset $A$ of $Y$ there is a finite ...
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1answer
22 views

Prove $x $ is not an element of $E^0$ if and only if $B_r(x) \cap E^c \neq \emptyset $ for all $r> 0$.

Prove: $x \notin E^0$ if and only if $B_r(x) \cap E^c \neq \emptyset $ for all $r> 0$. Proof: I just need help with converse part. Converse: Suppose $B_r(x) \cap E^c \neq \emptyset $ for all ...
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“$\sigma$-uniform continuity”

Let $X$ be an arbitrary metric space and $f:X\to\mathbb R$ a bounded continuous function. Is it possible to choose a countable sequence $(A_n)_{n\in\mathbb N}$ of (preferably open or closed) subsets ...
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Is there a metric proof for the Caristi theorem?

I'm writing a paper about hyperconvexity in metric spaces and came across Caristi's theorem: Let $(X, d)$ be a complete metric space and $\phi\colon X \to \mathbb{R}^+$ a continuous function. An ...
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23 views

discrete metric, both open and closed.

I've checked several answers though, still don't understand last bit. Taking radius r = 1/2 then every subset is singleton and it is open. But then how do you deduce it is also closed? Well, a ...
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1answer
32 views

Sequence in product metric space [on hold]

Let we have $(X_1,d_1)$ is a metric space and $(X_2,d_2)$ is another metric space . Now we will difend $X=X_1*X_2$ and we have $d$ is a distance function on $X$ So $(X,d)$ is a metric Space I ...
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metric spaces byE.T.copson solutions for exersice of chapter 4

M is the set of all analytic function of the complex variable zregular on the unit disc lzl<1 such that sup ( int 0<=r<1
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Show that there exists sets $A, B$ in $R$ such that $(A \cup B)^o \neq A^0 \cup B^o$

$\newcommand{\closure}{\operatorname{closure}}$ Show that there exists sets $A, B$ in $R$ such that 1) $(A \cup B)^\circ \neq A^\circ \cup B^\circ$ and $2)$ $\operatorname{closure}(A \cap B) \neq ...
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Closure of $A= (0,1) \cup (1,2)$ vs. Closure of $A = [0,1] \cup \{2\}$

Closure of $A= (0,1) \cup (1,2)$ vs. Closure of $A = [0,1] \cup \{2\}$ I am trying to figure out the difference of the closure of these two sets. Informally, my definition of closure is the ...
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1answer
23 views

Surjectivity of expanding map

Suppose that $(X, d)$ is a compact metric space and that $f: X \rightarrow X$ is a continuous function satisfying $d(x,y) \leq d(f(x), f(y))$ for all $x, y \in X$. Show that $f(X) = X$. Here is a ...
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1answer
25 views

Does $\sigma$ -compact imply separable?

Let $D$ be a metric space. If $D$ is $\sigma$-compact, does this imply that $D$ is separable? I thought I had a proof, but I think it is wrong. my proof: Let $K_n$ the compact sets such that $K_n ...
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Takahashi convex metric spaces

A Takahashi convex metric space is a metric space $(X,d)$ such that $\exists W : X \times X \times [0,1] \rightarrow X$ that satisfies : $d (u, W(x,y; \lambda)) \leq \lambda d(u,x) + (1- \lambda) ...
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1answer
124 views

Show $h:A \rightarrow B$ 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 $X = \cup_{n≥1}C_n$, be a space and assume that a map h : A → B is such that each ...
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1answer
60 views

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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348 views

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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1answer
48 views

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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37 views

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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1answer
36 views

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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1answer
41 views

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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1answer
35 views

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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1answer
48 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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$d_1(x,y)=|x-y|$ & $d_2(x,y)=|\frac1x-\frac1y|$ 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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1answer
59 views

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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28 views

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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1answer
24 views

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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1answer
19 views

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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3answers
106 views

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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1answer
28 views

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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69 views

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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0answers
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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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1answer
25 views

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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1answer
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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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1answer
33 views

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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1answer
30 views

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: ...