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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Regularity of measures proof

A probability measure $\mathbb P$ on a metric space $(S,d)$ is closed regular if $$ \mathbb P(A) = \sup \{ \mathbb P(F) : F \subseteq A, F \text{ - closed} \} \text{*}$$ with $A\in ...
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Hausdorff metric with Kuratowski sum

Let $(X, \rho)$ be a metric space. Let $A$, $B$, $C$ and $D$ be non-empty closed subsets of $X$. Let $h$ denote the Hausdorff metric. I need to show that $h(A+B, C+D) \leq h(A,C) + h(B,D)$ where ...
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how is this proof about distance in metric spaces is wrong?

Let $A,B \subseteq \mathbb{R}^d$ be non-empty sets. Define their distance to be $$ d(A,B) = \inf \{ ||x-y|| : x \in A, \; \; y \in B \} $$ For any $A,B$, do we have that $d(A,B) = d( \overline{A}, ...
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Example of a sequence function which is unbounded

Let $(X,d)$ be a metric space and $T:X\to X$ be self map on $X$. A map $\phi :X\to[0,\infty)$ is said to be sequence function with respect to $T$ if $$\lim_{n\to\infty}\phi(x_n)<\infty$$ whenever ...
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1answer
47 views

Does metrizability means only the inherit of topological properties?

A topological space X is metrizable if it is homeomorphic to a metric space. So, while it has a homeomorphism between X and a metric space, it means that they have same topological properties. My ...
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65 views

Why are isometries continuous?

Definitions Let $(X, \mathfrak{T}_X), (Y, \mathfrak{T}_Y)$ be two topological spaces and $f: X \rightarrow Y$ be a mapping. $f$ is called continuous $:\Leftrightarrow \forall U \in \mathfrak{T}_Y: ...
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61 views

When is distance to the boundary always less than that to the exterior?

Let $X$ be a metric space with the distance function $d$. Given a subset $S \subseteq X$, what are the required conditions on $X$ and $S$ are so that $d(x, \partial S) \leq d(x, \operatorname{ext} S)$ ...
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54 views

Why is pointwise continuity not useful in a general topological space?

On page 27 of Lee's Introduction to Topological Manifolds, he writes In metric spaces, one usually first defines what it means to be continuous at a point...in topological spaces, continuity at a ...
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Uniform convergence and pointwise convergence

Having the standard definitions for pointwise resp. uniform convergence of sequences of functions in a general metric space ($X,d$). What special conditions should X fulfill such that pointwise ...
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1answer
53 views

Does this claim depend on topology?

An open rectangle is a set $R\subset\mathbb{R}^2$ of the form $R=\{(x,y)\in\mathbb{R}^2;\;a<x<b\text{ and }c<y<d\}$, where $a,b,c,d\in\mathbb{R}$, $a<b$ and $c<d$. Let $\|\cdot\|$ ...
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Metrizable spaces

A topological space X is metrizable if it is homeomorfic to a metric space. I want to know does this mean that all of topological properties of a metric space inherit to that topological spaces? Also ...
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Cone metric spaces and fixed point theory

These days cone metric spaces, as a generalization of metric spaces, started to be very interesting... A special interest for this space is its application to the fixed point theory. On the last ...
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103 views

Clarification on trivial topology not being metrizable.

This is my understanding of the proof. Please let me know if I am correct. If I take any two points in a trivial topology. I can find the minimum radius such that they both are in two different open ...
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1answer
17 views

Metric on half-open interval s.t. subset is open w.r.t. $d$ iff open w.r.t. Euclidean metric

I wish to find a metric $d$ on the space $X = (0,1]$ such that $(X,d)$ is complete and so that a subset of $X$ is open with respect to $d$ if and only if it is open with respect to the Euclidean ...
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117 views

Difference between closed, bounded and compact sets. [closed]

Can somebody explain the difference between compact, bounded and closed sets with examples?
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28 views

Are a closed and a compact disjoint sets sufficiently wide apart?

Let $X$ be a metric space and $A$ and $B$ two subsets of $X$. If $A$ is closed, $B$ is a compact, and $A\cap B=\emptyset$, is it true that there is $d>0$ such that $\operatorname d (x,y)\ge d$ for ...
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87 views

Does Cantors Theorem imply one can never build nested closed sets with empty intersection?

Granted, I know Cantors Theorem says that it has to be nonempty, but what if we have some Cauchy sequence that is not complete? I was thinking it would be possible then.
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Do all subsets of metric spaces have boundry points?

I am learning about metric spaces. I failed to find an aunambiguous answer to my question on Google. So this is the right place to ask: Assume (X,d) is a metric space, and $A \subset X$. If I ...
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1answer
19 views

Can you give me an example of a function that is either upper OR lower quasi-continuous but not both?

A function $f: X \rightarrow \mathbb{R}$ is said to be upper (lower) quasi-continuous at $x \in X$ if for each $\epsilon >0$ and for each neighbourhood $U$ of $x$ there is a non-empty open set $G ...
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1answer
47 views

Prove that $\exists\ C$ such that $T: C \to C$ with $C=\overline{\text{conv}}T(C)$

I have a problem: Let $K$ be a nonempty, closed, bounded and convex subset of reflective Banach space $X$. Suppose that $$T:K \to K$$ is nonexpansive. Prove that $\exists\ C$ such that ...
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50 views

Isometries of a general metric

For a general (pseudo-)Riemannian manifold, i.e. in which the interval $ds$ can be written $ds^2 = g_{ab}\,dx^a \,dx^b$, is there a general prescription for finding the group of isometries- by ...
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1answer
69 views

When is $\mathcal C(X)$ complete?

I learned today in class that $\mathcal C([a,b])$ is complete with the supremum norm. That is, any uniformly convergent sequence $(f_n)$ is Cauchy, and the converse is true. I asked my teacher about ...
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Concerning $C^0[0,1]$ and the $L^1$-Norm.

Consider the well known Euler sequence of functions $x^n$ ($n=1,2,3\ldots$) on $[0,1]$. It is clear that it converges against $\chi_{1}$, the characteristic function of the singleton $1$, in the ...
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2answers
61 views

why the inequality property is important of metric space?

New for topology here. There is a property in the axiomation of metric space: $d(x,z)\le d(x,y)+d(y,z)$. I understand that property applies well when it come to the measure of distances among ...
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1answer
22 views

Metric space contractions [duplicate]

Consider $(\mathbb{C}^n ,||\cdot||_{1})$ where $||x||_{1} = \sum\limits_{i=1}^{n} |x_{i}|$ and $d_{1}(x,y) = ||x-y||_{1} = \sum\limits_{i=1}^n |x_{i} - y_{i}|$. I want to find a condition such that ...
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78 views

Can an open ball have just one point. As per my understanding it cannot. Please clarify.

I am new to functional analysis and am just learning. To my understanding an open ball must have at least 2 points else its definition will not be satisfied. Now if I have just an empty set and this ...
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1answer
40 views

How to prove that this set is open.

This is probably very easy. I know that it is very obvoius, but I want to prove it using the definition of what beeing open means. I have $A = \{y \in R: y <a \}$. And I want to show that it is ...
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44 views

Sufficient conditions for existence of injection from a metric space $M$ to $\mathbb{R}$

Let $M$ be any metric space. What conditions are required of $M$ for there to exist an injective, continuous function $$\varphi \colon M \longrightarrow \mathbb{R}$$ I would like to believe that ...
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3answers
30 views

On proving that a compact metric space is bounded

I am trying to prove the above claim. My question is on the bolded part and if I am correct in making that statement. A metric space is bounded if there exists an $M \in \mathbf{N}$ such that ...
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40 views

Showing that for equivalent metrics, the separability of one metric space implies the separability of the other

Since $\rho$ and $\sigma$ are equivalent, there are positive numbers $c_1$ and $c_2$ such that for all $x_1, x_2 \in X$, $$c_1\cdot\sigma(x_1, x_2) \leq \rho(x_1, x_2) \leq c_2\cdot\sigma(x_1, ...
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Finding a condition to make a map a contracting map

Consider $(\mathbb{C}^n ,||\cdot||_{1})$ where $||x||_{1} = \sum\limits_{i=1}^{n} |x_{i}|$ and $d_{1}(x,y) = ||x-y||_{1} = \sum\limits_{i=1}^n |x_{i} - y_{i}|$.
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41 views

Topology induced by $d(x,y) = \sum 2^{-i} \max \lbrace d_i(x_i,y_i) , 1 \rbrace$ on $\prod X_i$

Let $(X_i, d_i) $ be a sequence of metric spaces. $$d(x,y) = \sum_{i=1}^{\infty} 2^{-i} \max \lbrace d_i(x_i,y_i) , 1 \rbrace$$ is a distance on $\prod_{i \in \mathbb N} X_i.$ Is the topology induced ...
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Metric space, I would like to rewrite this proof, completeness and compactness

I am looking at metric spaces. For the metric space C(X,Y) the metric is: $\rho (f,g)=sup\{|f(x)-g(x)| :x \in X\}$ This is the proof I am reading about. I would like to do the proof without just ...
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1answer
56 views

Closure in [0,1]

If $\{p_1,..,p_n,...\}$ are the prime numbers. Let $\displaystyle A_n:= \left\{\frac{k}{p_n}\ : k\in\{1,...,p_n-1\}\right\}$ Which is the closure of $A\,=\displaystyle ...
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1answer
47 views

Connectedness and Continuity

Suppose $S^n\subset \mathbb{R}^n$ be the set of all points in $\mathbb{R}^n$ with unit Euclidean norm. That is $$S^n=\{x\in\mathbb{R}^n~:~\|x\|=1\}$$ If $f:S^{n-1}\rightarrow\mathbb{R},~n>2$, is ...
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26 views

Prove that every pseudocompact metric space is compact

This is from Real Mathematical Analysis by Pugh, problem 2.85(a). I've seen proofs but they've used concepts that haven't been covered up to this point, like the Tietze extension theorem, metrizable ...
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176 views

If every real-valued continuous function is bounded on $X$ (metric space), then $X$ is compact.

Let $X$ be a metric space. Prove that if every continuous function $f: X \rightarrow \mathbb{R}$ is bounded, then $X$ is compact. This has been asked before, but all the answers I have seen prove the ...
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2answers
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Openness w.r.t. these two metrics are equivalent.

Suppose $(X,d)$ is a metric space. Define $\delta:X\times X\rightarrow[0,\infty)$, as $$\delta(x,y)=\frac{d(x,y)}{1+d(x,y)}.$$ It is easy to show that $\delta$ is a metric as well, but I am having ...
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1answer
39 views

Questions about sum of sets

Suppose $A$ and $B$ are two subsets of $\mathbb{R}^n$. Then define $A+B=\{a+b~|~a\in A,b\in B\}$. I have proved that if $A$ and $B$ are open, then so is $A+B$. However, I need to prove that if $A$ ...
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Continuity from complete metric space

Let $f:X\rightarrow Y$ be a continuous function, such as: $f(X)=Y$. If $X$ is complete, does it imply $Y$ is complete?
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When does the quotient metric reduce to the infimum of the distances of only two points?

Given a metric space $X$ and an equivalence relation $\sim$, the quotient (pseudo-)metric on $X/\sim$ is defined as follows: $d'([x],[y]) = \inf \left \{ d(p_1,q_1) + d(p_2,q_2) + ... + ...
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1answer
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Set connected in the metric space.

If $X:=\{0\}\cup\{1/n\}_{n\in{\mathbb{N}}}$ (with the metric standard of $\mathbb{R}$) then my question is, a open ball with center in $0$ is connected? Thank you all. I'm confused
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How to complete this proof to show that the metric $d'(x,y) = d(x,y) / (1 + d(x,y))$ gives the same topology as $d(x,y)$ gives?

This is an exercise problem from Munkres's Topology (Exercise 11 of Section 20 "The Metric Topology", 2nd edition). Exercise 11: Show that if $d$ is a metric for $X$, then $$d'(x,y) = d(x,y) / (1 ...
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Does addition have to be defined in metric spaces?

I have some questions, my first question is: 1. Does addition have to be dfined in a matric space? I have some more quick questions, I will describe why I have them. Look at this excercise. I ...
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28 views

triangle inequality for metrics

I am trying to prove the triangle inequality for the space [0,1) and the metric $\rho$(x,y)= min{|x-y|,|x+1-y|, |y+1-x|}. Now, I know that I will need to separate it into several cases. The proof ...
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34 views

A basic question on converges in distribution

The distance $d(F,G)$ between two distribution functions is the infimum of those $\epsilon > 0$ such that $G(x-\epsilon) - \epsilon \leq F(x) \leq G(x+\epsilon) + \epsilon \quad\forall x $. Now I ...
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Convergence of $f_{n}(x)=\frac{1}{x^2+n^2}$ and $g_{n}(x)=\frac{2nx}{x^2+n^2}$ in sup norm

I need to show that (i) $f_{n}(x)=\frac{1}{x^2+n^2}$ converges to the zero function in sup norm, and (ii) $g_{n}(x)=\frac{2nx}{x^2+n^2}$ does not. Not sure if this is right but would appreciate ...
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Open sets are $\mu*$-measurable in a metric space with a given condition

I have a homework problem that I'm very stuck on. The problem statement is as follows: "Suppose that $X$ is a metric space, and that for any sets $E,F \subseteq X$, if dist$(E,F) > 0$ then ...
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60 views

Let $(X, d)$ be a metric space. Does a result proved for the metric $d$ apply to every metric $d^{'}$ on $X$?

Let $(X, d)$ be a metric space. Does a result proved for the metric $d$ apply to every metric $d^{'}$ on $X$ ? Very often a statement involving $\mathbb R^n$ is proved by applying the metric ...
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32 views

In a normed vector space, if $O $ is an open set then $ O+a $ is open?

In a normed vector space, if $O $ is an open set then $ O+a $ is open? Here $ a$ is an element of some other set $ A $ . This feels intuitively obvious, as we just have "moved" the entire set, but I ...