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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Compactness of $(a,b)$ in $\Bbb{R}$.

Say we have an open set $(a,b)\subset\Bbb{R}$, which has an infinite cover $\mathfrak{C}$. Let us assume $(a,b)$ is not compact. Then we can select an open set $(a_1,b_1)\subset (a,b)$ such that it is ...
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617 views

Cantor's intersection theorem for nested open sets.

Let $\{I_k\}$ be a set of nested closed bounded sets such that $\forall k\in\Bbb{N}, I_k\supset I_{k+1}$, and $\lim_{n\to\infty}\operatorname{diam}(I_n)=0$. Let the metric space $(X,d)$ be complete. ...
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64 views

Investigating the importance of boundedness in Heine-Borel's theorem.

We know that $[1,\infty)$ is not compact in $\Bbb{R}$. A simple proof: the cover $\mathfrak{C}=\bigcup B(n,\frac{3}{4})$ for all $n\in\Bbb{N}$ of $[1,\infty)$ does not have a finite subcover. What ...
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98 views

Open Subgroup of (R,+)

Let G be an open subgroup of (R,+) Show that G=R. Note: I've tried taking an interior point of G. Can Archimedian Property be used?
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314 views

metric property in a group

Can we define a metric function on a group $G$? Please give examples other than $\mathbb{R}$. Actually most groups have elements in discrete manner. It sounds vague but I can't be more precise.
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1answer
66 views

Prove that the function $d_A:X\to\mathbb R:x\mapsto\displaystyle\inf_{a\in A} d(x,a)$ is continuous.

For a metric space $(X,d)$ and a nonempty subset $A$ of $X$ prove that the function $$d_A:X\to\mathbb R:x\mapsto\displaystyle\inf_{a\in A} d(x,a)$$ is continuous. Choose $c\in X$ and ...
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139 views

When discussing compactness, is it necessary to specify the metric space?

My textbook has this question. "Show that (0,1] is not compact by finding an open cover with no finite subcover." It did not explicitly specify the metric space, so I started wondering, what if we ...
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1answer
484 views

A product of two sequentially compact metric spaces is compact. How to prove this explicitly?

We know that a product of two (or finitely many) compact topological spaces is compact. And we also know that in a metric space, compactness is equivalent to sequential compactness. So a product of ...
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222 views

Show that $C(X,\Bbb R)$ is not separable

Let $X$ be a metric space whose points are the positive integers and whose metric ($d\colon X\times X\rightarrow \mathbb{R}$) is defined by $$d(x,y)=\dfrac{|x-y|}{1+|x-y|}.$$ Show that ...
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1answer
129 views

Which topology book gives most complete account of the Hausdorff metric and similar other ones?

Which book (on topology) gives the most complete, yet accessible, account of the Hausdorff metric? the fuzzy metric? the cone metric? the probablistic metric? and so on? Somebody once gave me a ...
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63 views

Continuity of $f:x\mapsto\displaystyle\inf_{k\in\Lambda}f_k(x)$ [closed]

Let $(X,d)$ be a metric space and for $k\in\Lambda $, let $f_k:X\to\mathbb R$ be continuous where $f_k\ge 0$ for all $k\in\Lambda $. Is $$f:x\mapsto\displaystyle\inf_{k\in\Lambda}f_k(x)$$ continuous?
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the conditions are not sufficient to ensure that the map $d:X\times X \to\mathbb{R}$ is a metric on the set $X$

Show that the conditions: (i) $d(x,y)=0$ iff $x=y(x,y\in X)$ and (ii) $d(x,z)\le d(x,y)+ d(y,z), \forall x,y,z\in X$ are not sufficient to ensure that the map $d:X\times X \to\mathbb{R}$ is a ...
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139 views

Distance Between Subsets in Connected Spaces

Suppose $\langle X, d \rangle$ is a metric space. For any two sets $F,G \subseteq X$, by abuse of notation define $d(F,G) = \inf \{ d(f,g): f \in F, g \in G \}$. Let $\rho > 0$, $x \in X$, and ...
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732 views

Intersection of a closed set and compact set is compact [duplicate]

I've been stuck on the following problem for several days: Let $(M,d)$ be an arbitrary metric space and $S, T$ be subsets of $M$. If $S$ is closed and $T$ is compact, then $S \cap T$ is compact. I ...
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213 views

How well can we embed graphs with shortest path metric into $\mathbb{R}^2$ with Euclidean metric?

If we take the integer lattice in $\mathbb{R}^2$ and make edges from $(m,n)$ to $(m+1,n)$ and $(m,n+1)$, you get your typical city block street layout, and if we put the shortest path metric on the ...
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42 views

I've show that a is true. How to show that rests are false?

I've show that a is true. How to show that rests are false?
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38 views

can I say $\mathfrak{T_1}\subseteq\mathfrak{T}_2$?

$(X,d)$ be a metric space and define $p(x,y)=\min\{{1\over 2},d(x,y)\}$ $\mathfrak{T_1},\mathfrak{T}_2$ be the topology generated by $d,p$, I need to know relations between them. Well, open sets of ...
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156 views

Terminology for metric space with “anti-symmetric” distance

I'm interested in spaces that have a two-place function $d$ with non-negative real values, satisfying the following three conditions (for all $x$, $y$, $z$): $d(x, x) = 0$ $d(x, y) + d(y, z) \geq ...
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1answer
141 views

Inconsistency between two definitions of closure.

I'm currently taking a course on analysis over $\mathbb{R}^n$ and the book used defines that a point $p \in \mathbb{R}^n$ is a limit point of a set $X\subset \mathbb{R}^n$ if $p$ is the limit of some ...
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5answers
316 views

Let $(X,d)$ be a compact metric space. Prove that there exists a number $K$ such that $d(x,y)\leq K$ for each $x,y\in X$.

I'm reading Intro to Topology by Mendelson. The problem statement is in the title. My attempt at the proof is: Since $X$ is a compact metric space, for each $n\in\mathbb{N}$, there exists ...
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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 ...
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115 views

What exactly is $\lim_{n\to\infty}a_n$?

Is $\lim_{n\to\infty}a_n$ a term of the cauchy sequence $\{a_i\}$, or the limit? I suppose it can't be both. I am leaning towards limit because if we select any $a_k\in\{a_i\}$, we have ...
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1answer
65 views

Clarification of proof on the completion of a metric space using Cauchy sequences

This is in reference to the proof of the completion theorem of metric spaces. (To protect against link rot, here is a copy of the document being referenced: page 1, page 2, page 3) A proof is ...
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177 views

$O(n,\mathbb R)$ of all orthogonal matrices is a closed subset of $M(n,\mathbb R).$

Let $M(n,\mathbb R)$ be endowed with the norm $(a_{ij})_{n\times n}\mapsto\sqrt{\sum_{i,j}|a_{ij}|^2}.$ Then the set $O(n,\mathbb R)$ of all orthogonal matrices is a closed subset of $M(n,\mathbb ...
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676 views

Are there any connections between metric space and inner product space

As mentioned in title, are the any connections between inner product space(well, here we talk about only real space) and metric space? I kind of notice that the axioms satisfied by both inner product ...
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449 views

Is this proof that every convergent sequence is bounded correct?

I've tried the following proof that a convergent sequence is bounded but I'm not sure if it is correct or not. Let $(M,d)$ be a metric space and suppose $(x_k)$ is a sequence of points of $M$ that ...
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120 views

A function that is not contractive with respect to any metric

I am struggling with this homework question with is related to iterated function system and fixed point theory. The question is: Let $\Delta \in R^2$ be a filled non-degenerate triangle with ...
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791 views

Proving that $X$ is complete if $A\subset X$ is dense and every cauchy sequence in $A$ converges to a point in $X$.

I am having trouble proving the following statement: "Let $(X,d)$ be a metric space and $A$ a dense subset such that every cauchy sequence in $A$ converges to a point in $X$. Prove that $X$ is ...
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45 views

Seeking a clarification regarding the bijectivity of the mapping between isometric spaces.

If $X$ and $Y$ are isometric spaces, does the mapping between them $f:X\to Y$ have to be bijective? I feel only injectivity is required to satisfy the relation $$d_y(f(a),f(b))=d_x(a,b)$$ and ...
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462 views

Proving the completeness theorem of metric spaces.

I have to prove that every metric space is isometric to a dense subset of a complete metric space. My proof: Let $X$ be the metric space, and $\{p\}$ the set of limits of all the cauchy sequences in ...
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350 views

Is Cantor's Intersection theorem valid for only a countable number of nested closed sets?

Cantor's Intersection theorem says if $F_{n+1}\subset F_n$ $\forall n\in\Bbb{N}$, then $\bigcap_{i=1}^{\infty}F_i\neq\emptyset$. This is valid only in complete metric spaces, and the proof is based ...
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118 views

Proof involving a vacuously true statement

Let $S$ be a finite subset of a metric space. Show that it is closed. I know a set is closed if and only if it contains all of its accumulation points. Let $x$ be an accumulation point of $S$. I want ...
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Space of continuous functions with compact support dense in space of continuous functions vanishing at infinity

How can we prove that the space of continuous functions with compact support is dense in the space of continuous functions that vanish at infinity?
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546 views

Proving that if $X$ is a complete metric space and $A\subset X$ is nowhere dense in $X$, then there is an open set in $X$ disjoint with $A$.

Let $A\subset X$ be a nowhere dense set, where $X$ is a complete metric space. My book says there is an open set $S$ of radius less than $1$ such that $S$ is disjoint with $A$. I'm confused as to ...
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646 views

The proof of Cantor's Intersection Theorem on nested compact sets

The book "Metric Spaces" by Babu Ram says this about the proof of Cantor's Intersection Theorem: Create nested intervals $F_{n+1}\subset F_n$ such that $\lim_{n\to\infty}\text{diameter}(F_n)=0$. ...
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1answer
71 views

Show that for $n\in\mathbb N$ the mapping $f:M(n,\mathbb R)\to M(n,\mathbb R):A\mapsto A^n$ is continuous.

Show that for $n\in\mathbb N$ the mapping $f:M(n,\mathbb R)\to M(n,\mathbb R):A\mapsto A^n$ is continuous. ($M(n,\mathbb R)$ is identified with $\mathbb R^{n^2}$ as a normed liner space.) ...
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1answer
34 views

What is $d(x,F_k)$, where $F_k\subset \Bbb{R}$?

Let $F_k\subset \Bbb{R}$ be an open interval in $\Bbb{R}$, and $x\in \Bbb{R}$ a point. How is $d(x,F_k)$ defined? I came across this notation in my textbook and it is confusing me. Is ...
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253 views

Is the arbitrary product of metric spaces a metric space?

If $I_n = \{i \in \mathbb{N} : 1 \leq i \leq n\}$ and if $\mathcal{X}_n=\{(X_i,d_i) : i \in I_n\}$ is a finite family of metric spaces, we know that we can make their product $X = \prod_{i \in ...
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1answer
52 views

Confusion: “if $x\in X\setminus A$ is the limit point of $\{x_i\}$ in $A$, then it is not necessarily true that $x$ is a limit point of $A$.”

Let $X$ be a metric space, and $A\subset X$. I read that if $x\in X\setminus A$ is the limit point of $\{x_i\}$ in $A$, then it is not necessarily true that $x$ is a limit point of $A$. How is this ...
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160 views

Prove equivalent metric spaces

Let $X_1=[1,2]$ and $X_2=[0,1]$. Let $d_1$ denote Euclidean and let $d_2(x,y)=2|x-y|$ in $X_2$. Show that $(X_1,d_1)$ and $(X_2,d_2)$ are equivalent metric spaces. How do I do that?
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198 views

The algebra of clopen sets vs. the algebra of connected components

Let $X$ be a topological space which, for my intents, may be assumed to be metrizable and compact if needed (let's say it's a closed subset of the unit cube or something like that). I know that: If ...
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1answer
33 views

Isolated points in $B(S)$?

Let $B(S)$ be the set of all bounded real-valued functions on $S$ where $S$ is any non-empty set. Let the metric be the metric of uniform convergence. Does $B(S)$ have any isolated points if $S$ is ...
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1answer
387 views

Recognizing uppersemicontinuous function as a pointwise decreasing limit.

Let $X$ be a compact metric space and $f:X\rightarrow \mathbb{R}$ be upper semicontinuous. Then why is it that $f$ is the pointwise decreasing limit of continuous functions? My attempt has been to ...
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Compactness used to get a covering by special smaller balls

Suppose $(X,d)$ is a compact metric space. Suppose we have a set $A \subseteq X$ such that the set of open $\epsilon$-balls around the points of $A$ cover $X$. I've read that "By compactness, there ...
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1answer
225 views

Bounded sets of isolated points in compact metric spaces

Context and definitions: Say $(X,d)$ is a compact metric space, with $f: X \rightarrow X$ continuous. For each $n \in \mathbb{N}$, the metric $d_{n}(x,y) = \max_{0 \leq k \leq ...
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1answer
112 views

Set of all matrix of rank $ r $ is open set in $ M_n (\mathbb { R })$

I have no idea how to start it. Actually I have no idea which matrix in $ M_n (\mathbb {R})$ are of rank $ r $. I know all basic result about it. please help me.
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53 views

Is $B(x,r)\cap A$ a ball in $X?$

I know that for a metric space $(X,d)$ and $\emptyset\ne A\subset X$ if we consider $A$ as a submetric space induced by $d$ then for any ball $B_A(x,r)$ in $A,$ $B_A(x,r)=B(x,r)\cap A.$ Now is it true ...
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1answer
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If every convergent subsequece converges to the same limit then the sequence converges

I came across this question: In a compact metric space $(X,d)$ if every convergent subsequence of a sequence converges to the same limit, say $l$, then the original sequence also converges to $l$.
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420 views

Is it true that any metric on a finite set is the discrete metric?

Is it true that any metric on a finite set is the discrete metric? I can see that it's at least equivalent with the discrete metric since $B(x,\delta)=\{x\}$ where $X=\{a_i\}_{i=1}^n,$ ...
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1answer
328 views

Metric vs metrizable spaces

A. Helemskii in the book "Lectures on functional analysis" write (in my horrible translation): The category of Hausdorff topological spaces (morphisms are continuous maps) contain the full ...