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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Convergence and metric - Proof?

Let $(x_n)$, $(y_n)$ be two sequences in a metric space $(P,d)$. Suppose $(x_n)$ converges to $x$ and $(y_n)$ converges to $y$. Prove that $\displaystyle\lim_{n \to \infty} d(x_n,y_n) = d(x,y)$ My ...
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16 views

Subsets of a metric space in which Hausdorff semi-distance is symmetric

These are the definition of Hausdorff distance and Hausdorff semi-distance for subsets of a metric space $X$. ‎‎Hausdorff semi-distance of two subsets ‎$‎A‎, B‎ \subset X$ is defined as below: ‎$‎d(A ...
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Sequential Equivalence Implies Topological Equivalence

Define two metric spaces $(M,d)$ and $(M,\rho)$ to be equivalent, denoted $d\sim p$, to mean that: Topological Definition $\forall x\in M: \forall \epsilon>0 \exists \delta_1>0, \delta_2>0: ...
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if $A$ is open in $M$ and $B$ is open in $N$, then $A \times B$ is open in $M \times N$

where $d((m_1,n_1),(m_2,n_2)) = d_M(m_1,m_2) + d_N(n_1,n_2)$ By some propositions, $A$ is open in $M$ if there exist an open set $K_1$ such that $A = M \cap K_1$ Also, there exist an open set $K_2$ ...
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38 views

Prove the following is a metric space…

I need to prove the following is a metric space over the integers: $b \geq 2$. For distinct integers $x, y$. Let $N(x,y)$ be the greatest integer $n$ such that $b^n$ divides $(x - y)$. Let $d(x,y) = ...
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24 views

What does this function converge to in $\mathbb{R}$ equipped with discrete metric?

We're given this function $f_n (x) = \begin{cases} 0 \ \mbox{ if $x <1/n$}\\ 1 \ \mbox{ if $x \geq 1/n$} \end{cases}$ I think it converges pointwise to $f(x) = \begin{cases} 0 \ \mbox{ if $x ...
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35 views

Pointwise convergence implies uniform convergence in discrete metric space $(X,d)$?

In general, does this hold for a sequence of functions in an arbitrary $X$? For a sequence to converge in the discrete metric, the sequence needs to become a constant sequence for a sufficiently large ...
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42 views

Open set in subspace not open in the entire space example

I am stuck with the following problem: X is a metric space. Suppose that Y is a subspace of X. Give an example that an open set in Y is not open in X. My own approach was this: Suppose U is a subset ...
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32 views

Convergence and finer topology

Can convergent of sequence be used to determine which topology is finer(in general topological space). I am asking this is question in effect of theorem on metric space: 'topology 1 is finer than ...
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31 views

Convergence of sequence and interior points

For a subset $A \subseteq X$, consider the statement, "$x$ is an interior point of $A$ iff for every sequence $(x_m)$ in $X$ converging to $x$ there exists $n \in \mathbb{N}$ such that for all $m > ...
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41 views

Connected Sets on Metric Spaces

I'm taking a first course in real analysis, and we're using Rudin's Principles of Mathematical Analysis as our main (only) book. In chapter two, Rudin discusses basic topology from the point of view ...
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46 views

Distance between a point and a closed set in metric space

Here is what I am thinking. Let (X,d) be a metric space and let C be a closed subset of X. Fix any poin p in X. Then, there exists a point q in C such that d(p,q) = distance(p,C). I think this ...
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51 views

Example of a locally compact metric space whose completion is not locally compact

Can someone suggest an example of a locally compact metric space whose completion is not locally compact?
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30 views

What are the epis in Met?

I have an assignment to precisely describe epimorphisms and monomorphisms in Met (category whose objects are Metric spaces and whose morphisms are contractions). I have shown that Mono $\iff$ ...
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Point-set topology - derived sets, boundaries, etc.

Consider the metric space $(\mathbb R^2,\sigma)$, where $\sigma$ is the discrete metric. Let $$E=\lbrace(x,y)\in \mathbb R^2 : x^2+y^2<1 \rbrace$$ i.e. the unit disc. Find the following sets ...
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$S^1$ with length metric is not isometric to any subset of Euclidean plane (metric given by restriction)

Let $S^1$ denote point whose radius is 1 from the center. Metric is given by distance between two point is the shortest distance, that is the length metric. Prove that $S^1$ with this metric is not ...
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if $\Bbb B=\{x\in \Bbb R^{n+1}; \langle x,x\rangle<1\}$ be a open ball from Euclidean Space $\Bbb R^n$

I study Metric spaces and I has this problem Show that sphere $\Bbb S^n=\{x\in \Bbb R^{n+1}; \langle x,x\rangle=1\}$ is metrically homogeneous. For the other hands, if $\Bbb B=\{x\in \Bbb R^{n+1}; ...
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1answer
15 views

Is Cantor set $F_{\sigma}$ set or $G_{\delta}$ set?

Is Cantor set an $F_{\sigma}$ set? or a $G_{\delta}$ set? There are similar questions on stackexchange, which consider a subset of Cantor set. But, I don't find the question posted above.
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30 views

Where was I supposed to use compactness in this proof that a compact subspace can be separated from a point by open sets?

Can anyone check my proof below? P. Let $X$ be a metric space. Prove that if $K\subseteq X$ is compact and $x\notin K$, there exist disjoint open sets $U$ and $V$ such that $K\subseteq U$ and ...
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Doubts about definition of open sets in “Understanding Analysis” by Stephen Abbott

In the book "Understanding Analysis" by Stephen Abbott, the author defines an open set as: A set $O \subseteq \mathbb{R}$ is open if for all points $a \in O$ there exists an ...
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Compactness of a set of bounded functions in the uniform norm

Let $T$ be a non-degenerate compact interval in $\mathbb R$ and $f:\mathbb R^2\to\mathbb R$ a strictly concave function such that (a) $f(0,0)=0$, (b) $f$ strictly increases in the first argument, and ...
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Give an example of A continuous function from X onto Y where X=[0,1] ;Y=[0,1]×[0,1] [duplicate]

Give an example of A continuous onto function $f:X\to Y$ where $X=[0,1]$ ;$Y=[0,1] \times[0,1]$.Why can't this function be one to one on [0,1]? As far as an example is concerned the only way to ...
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Is there a reason why M can't be all summable sequence?

Let M be the set of all summable non-negative sequences $\{x_k\}_{k=1}^\infty$ of real numbers, that is, $x_k \geq 0$ for all k and $\sum_{k=1}^\infty x_k$ converges to a real number. Let $d:M \to ...
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66 views

Show that $\operatorname{diam}(A\cup B)\le \operatorname{diam}(A)+\operatorname{diam}(B)+d(A,B)$

I'm beginning to study metric spaces and I see this question Consider $A$ and $B$ bounded and non-empty subsets of $M$, where $M$ is a metric space. Show that $\operatorname{diam}(A\cup B)\le ...
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What is $d(\sin(x),\cos(x))$ if d is a distance function in a metric space?

Let $M=\{f:[a,b] \to \textbf{R} | f \,is \,continuous \}$. Let $d:M \to \textbf{R}$ be defined by $d(f,g)=\int_a^b |f(x)-g(x)| \,dx$. What is d represent geometrically, and show that M, d is a metric ...
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49 views

Point-set topology, characterizing sets

I think the question sort of speaks for itself. Please no solutions in the answers - I'm mostly looking to see if my logic makes sense: Let $E$ be a set in a metric space $(X,d)$. Let $E^{\circ}$ ...
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19 views

The measures used to define Hausdorf dimension versus Haar measure

I don't know if there is a name for the measure, so let me construct it: Given a metric space $X$ we define the outer measures $H_\delta^\alpha$ by $\forall A \subset X$ $H_\delta^\alpha ...
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Problem 1, Chapter 3 - A Comprehensive Introduction to Differential Geometry - Spivak

There seems to be an error in item (d) of Problem 1, Chapter 3 of Spivak's A Comprehensive Introduction to Differential Geometry, 3rd Edition, which I transcribed below. There is something wrong with ...
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Product metric spaces is again a metric space

Let $(X,d_X)$ and $(Y,d_Y)$ be metric spaces, and let: $$ d_2 ((x_1,y_1),(x_2,y_2)) = \left[d_X(x_1,x_2)^2 + d_Y (y_1,y_2)^2 \right]^{\frac{1}{2}} $$ for the points $(x_1,y_1)$ and $(x_2,y_2)$ in $X ...
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Mapping on induced topology and distance metric

Let $(X, d)$ be a metric space. Let $τ$ be the metric topology on $X$ induced by $d$. For $A ⊆ X$ , let $d(x, A) := \inf_{a∈A} d(x, a) $ for $x ∈ X$ (a) If $f (x) := d(x, A)$ (for a fixed subset ...
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67 views

Relative compactness of metric space

I know that in a metric space $X$ compactness, countable compactness and sequential compactness of a subspace $X'$ are equivalent using the definition of countable compactness as every infinite subset ...
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1answer
86 views

Relation between the covers by sets of small diameter and the size of uniformly separated sets

Sorry I didn't find a better title. Here is the problem and my solution so far, I'd appreciate if someone could told me if is correct and for the last point, which at first sight seems to be ...
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Distance between point and set

For a non-empty subset $A$ of $\mathbb{R}^n$, and any $x\in \mathbb{R}^n$, define $d(x,A)=\inf\{ |x-a|\colon a\in A\}$. The problem is to show that if $A$ is closed and for any $r>0$, the set ...
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1answer
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Is there any metric $d$ of $\mathbb R^n$, $n<\infty$ such that $\mathbb R^n$ is bicompact and no norm induces $d$

There are some simple metrics can't yielded by norm .But add bicompact,I can't structure such example. In fact ,I want to know the condition of metric can be yielded by norm. Sorry for my poor ...
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1answer
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Is it true that $ d(B _{n+1 } ,A _n ) \ge \frac {1 } {n (n+1) }$?

Define $A _n = \{x \in F ^c \cap A: d(x,F)\ge 1/n \} $, where $F $ is a closed subset, and $A$ any subset of a metric space $X $. Then let $B _n =A _{n+1 } \cap (A _n ) ^c$ I have two questions: 1) ...
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1answer
28 views

Limit points and boundary points of a general metric space

Let X be a metric space, E a subset of X, and x a boundary point of E. It is clear that if x is not in E, it is a limit point of E. Similarly, if x is in E, it is a limit point of X\E. And there ...
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Definition of neighborhood

I am starting to work through Rudin's Principles of Mathematical Analysis. For $(X, d)$ a metric space and $x \in X$, Rudin defines the neighborhood $N_r(x)$ of $x$ to be the set consisting of all ...
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60 views

how to find a metric to make a space complete (help)

Hi everyone I'm struggle with the following. Define a complete metric on $\mathbb{R}\setminus \{0,1\}$ with usual relative topology. I'd like to follow the big hint of Daniel Fischer but I have ...
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Completeness of $C_{X,\mathbb{C}}$

If I haven't committed any error in my proof, the space of continous applications mapping a compact space $X$ into $\mathbb{C}$ or $\mathbb{K}$ is complete with the metric defined by ...
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Proof for distances to a set

With a metric space $(X,d)$, prove that $|d_E(x)-d_E(y)|\leq d(x,y)+d(y,z)$. In this context, $x \in X$, $d_E(x)=\inf\left\{d(x,z) : z \in E\right\}$, E is a subset of X. I've already proved the ...
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26 views

Finite intersection property in any metric space

If $(X,d)$ is any metric space and $\{A_\alpha\}_{\alpha\in I}$ is a collection of nonempty compact subsets of $X$ such that the intersection of any finite subcollection of sets is non empty does that ...
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$\epsilon$-isometry of a compact metric space is $\epsilon$-surjective

The question whether an isometric map $f : X \to X$ of a compact metric space is surjective has been asked (and answered positively) frequently. Assume more generally that $\vert d(f(x),f(y)) - ...
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A map $f:([a,b], |\cdot|) \to ([c,d], |\cdot|)$ is an isometry if and only if $d-c = b-a$.

I was asked to prove the following problem: A map $f:([a,b], |\cdot|) \to ([c,d], |\cdot|)$ is an isometry if and only if $d-c = b-a$. But I think this is not correct, specifically the sufficient ...
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$d$ is a metric space on $X\not=\{0\}$, obtained from a norm. $d'(x,y)=d(x,y)+1$. Show $d'$ cannot be obtained from a norm.

If $d$ is a metric on a vector space $X\not=\{0\}$ which is obtained from a norm, and $d'$ is defined by $d'(x,x)=0$, $d'(x,y)=d(x,y)+1, (x\not=y)$, show that $d'$ cannot be obtained from a norm. ...
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triangle inequality for a metric space

If $d_{\infty}(a,b) =$ max$\{|a_{i} - b_{i}|\}$ for $1 \leq i \leq k$, I want to prove that this is a metric on $\mathbb{R}^k$. Its pretty clear that $d_{\infty}(a,a) = 0$ and it is also pretty clear ...
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Metric on natural numbers united with infinity

can anyone give me an example for the following metric $d$? Let $\Omega = \mathbb{N}_+ \cup \{ \infty \}$ and $d$ be a metric such that all points $n \in \mathbb{N}_+$ are isolated w.r.t. $d$ and ...
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Inner product spaces, normed spaces, metric spaces and topological spaces

I am collecting theorems or properties that hold in IPS, NS, MS or topological spaces, but not all of them. The reason is that I want to create some sort of overview over the respective spaces and ...
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1answer
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Does uniform convergence depends on the metric?

Definition: Let $f_n:X\to Y$ be a sequence of functions from a set $X$ to the metric space $Y$. Let $d$ be the metric for $Y$. The sequence $(f_n)$ d-converges uniformly to the function $f:X\to Y$ if ...
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34 views

Assumptions of Kolmogorov extension theorem

As far as I know, a class of spaces for which the Kolmogorov theorem works and which is closed under countable products, are the spaces of complete separable metric spaces which are also called Polish ...
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Prove that the space of sequences under this metric is complete and compact.

I'm currently studying for the prelim exams, and I would love a hint on how to complete this problem. If $X$ is the space of sequences of $0$'s and $1$'s (i.e., $x \in X$ if $x = (x_{1}, x_{2}, ...