Metric spaces are sets on which a metric is defined. A metric is a generalisation of the concept of "distance". Metric spaces should not be confused with topological spaces.

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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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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 $\iff$ one-to-...
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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 [note ...
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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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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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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 $x\...
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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 $\epsilon$-...
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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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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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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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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 (A)=\inf\...
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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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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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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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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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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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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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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 $d(f,g):=\sup_{x\...
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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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Completeness and Separability of $C[0,\infty]$

Let $C[0,\infty]$ be the space of all continuous functions on $[0, \infty ]$ with metric $$ \phi(\omega_1, \omega_2) = \sum^{\infty}_{n=1} \frac{1}{2^n}\max_{0{\leq} t {\leq} n}(|\omega_1(t)-\...
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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)) - d(x,...
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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. I'...
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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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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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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}, x_{...
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Why is the triangle inequality property of a metric space important? [duplicate]

From my understanding, when we use metric spaces, we are trying to measure how "different" certain elements in a metric space are from one another. We all know that a metric space $(S,d)$ satisfies: ...
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Does every non-compact bounded metric space support an equivalent metric in which it is unbounded?

Consider $X$ be an infinite set. Let $d$ be a non compact bounded metric on $X$. Can we define an unbounded metric $d'$ on $X$ such that both the metric spaces $(X,d)$ and $(X,d')$ give the same ...
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For continuous functions, preimage of open set is open.

Let $f$ be a continuous function from a metric space $X$ into $Y$. If $V\subset Y$ and $V$ is open, then show that $f^{-1}(V)$ is open. The proofs I've seen of the fact that open sets have open ...
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Continuous function vs Uniformly continuous function

Can you give me an example of the function in metric space which is continuous but not uniformly continuous. Definitions are almost the same for both terms. This is what I found on wiki: ''The ...
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No direct proofs of “if $ f: (X, d_X) \to (Y, d_Y)$ is continuous and $X$ is compact then $f$ is uniformly continuous.”

I am studying the theorem "if $f:(X,d_X)\to (Y,d_Y)$ is continuous and $X$ is compact, then $f$ is uniformly continuous." I am not looking for a proof, but I have an argument against any attempt at a ...
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Reference for convergence properties of the Hausdorff distance

Consider the following properties of the Hausdorff distance in $\mathbb R^n$. Let $\Omega_n \supset \Omega_{n+1} \supset ...$ a sequence of open, convex and bounded sets with $\operatorname{int}(\...
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The metric identification of a pseudometric on $C(\mathbb{I})$

I have a pseudometric $\mu$ on $C(\mathbb{I})$ defined by $$\mu(f, g) = |f(x_0) - g(x_0)|.$$ I then take the metric identification of $(M, \mu)$ and am asked what familiar space this metric ...
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Every $K$-Lipschitz function can be uniformly approximated by $C^1$ functions with derivative bounded by $K$

The exercise states: Let $a,b\in\mathbb{R}$, $a<b$ and let $(C[a,b],\Vert\cdot\Vert)$ denote the vector space of continuous real functions on $[a,b]$ endowed with the uniform norm. Let $C^1[a,b]...
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Mary Ellen Rudin's proof that all metric space are paracompact

Given a metric space $(X,d)$, show that the space is paracompact. I have no idea where to begin on this, and the proofs of this I have seen have been difficult for me to understand. Can anyone offer a ...
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Let $(X,d)$ be a metric space and $M\subset X$ is uncountable. There is $\alpha>0$ such that $d(x,y)=\alpha$. Prove that $X$ is not separable.

Let $(X,d)$ be a metric space and $M\subset X$. Suppose $M$ is uncountable and there is $\alpha>0$ such that $d(x,y)=\alpha$ for every $x,y\in M$ with $x\not=y$. Prove that $X$ is not separable. ...
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Problem showing that $\partial D = \emptyset$

I am considering $(X,d)$, which is a set with a discrete metric and $D \subset X$. With these conditions given, I am supposed to show that $\partial D = \emptyset $. Further I am supposed to describe ...
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closure = union of the set and the set of limit points

Let $S \subseteq \mathbb{R^n}$ and denote the set of limit points of $S$ as $S'$. Show that $S \cup S' = \bar{S}$ where $\bar{S}$ is the closure. (i) I want to show that $\bar{S} \subseteq S \...