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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If $E \subset\mathbb R$ is compact and nonempty. Prove $\sup (E)$, $\inf (E) \in E$

Suppose that $E \subset \mathbb R$ is compact and nonempty. Prove $\sup (E)$, $\inf (E)\in E$. attempt: Suppose $E$ is compact, then $E$ is closed and bounded. Thus $\sup(E)$ and $\inf (E)$ exist. ...
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47 views

Let $A,B$ be compact subsets of $X$. Prove that $A \cap B$ is compact.

Let $A,B$ be compact subsets of $X$. Prove that $A \cap B$ is compact. Attempt: Suppose by contrapositive, that $A \cup B$ is compact. Then let $V$ be an open cover of $A \cup B$. Then let $A$ be ...
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a connect set in plane [on hold]

Let $$E=\{(x,y)\in\Bbb{R^2}:x\in\Bbb{Q^c}\,\ or\,\,y\in\Bbb{Q^c}\}$$ show that $E$ is connect,is $$E=\{(x,y)\in\Bbb{R^2}:x\in\Bbb{Q}\,\ or\,\,y\in\Bbb{Q}\}$$ connect?
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Set that is bounded but not totally bounded: Reading textbook

I've been reading a Real Analysis textbook that my friend loaned to me. I have come across a proposition that says that a totally bounded set is bounded, but a bounded set is not always totally ...
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31 views

Prove there exists $a \in E$ such that $a = f(a)$, assuming $d(f(x), f(y)) \le Kd(x,y)$ with $K<1$

Let $f: E \rightarrow E$, $E$ a complete metric space. Assume that there exists $K$ such that $0 < K < 1$ and $d(f(x), f(y)) \le Kd(x,y)$ for all $x,y \in E$. Prove that there exists $a \in E$ ...
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1answer
20 views

Criterion for Isometry

Let $X$ be a topological vector space, with $d$ an invariant metric compatible with the metric. Let $f:X\to X$ be an involutive linear isomorphism. How do you show that $f$ is an isometry? I ...
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25 views

Prove a sequentially compact metric space is bounded.

Prove that if the metric space $(X, d)$ is sequentially compact, that there exists points $x_0$ and $y_0$ belonging to $X$ such that; $$d(x, y) \leq d(x_0, y_0)$$ for every $x$ and $y$ belonging to ...
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21 views

On convergent sequences

Suppose that i have and open and surjective map between two metric spaces $\pi\colon X\to Y,$ and a sequence $(x_n)_{n\in \mathbb{N}}$ such that its image by $\pi$ converges. Is it true that ...
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Intersection of balls in Hamming space

Let $B(x_1, r)$ and $B(x_2,r)$ be balls in $\{0,1\}^n$ (in Hamming distance). Denote by $d$ Hamming distance between $x_1$ and $x_2$. What is $|B(x_1, r) \cap B(x_2, r)|$ (asymptotically)? Upd: I ...
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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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81 views

$X$ 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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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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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
23 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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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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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 ...
2
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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 $2)$ $\operatorname{closure}(A \cap B) \neq ...
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“$\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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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
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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
27 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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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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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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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
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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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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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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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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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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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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
36 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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61 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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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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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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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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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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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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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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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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1answer
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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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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
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: ...
2
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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 ...
8
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1answer
99 views

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 ...