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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What to do after defining a metric on a set? [closed]

Given a finite set $M$ of binary sequences of length 6: $$ M=\{\{1,0,1,0,0,1\},\{1,0,0,0,1,1\},...\} $$ Let's define a metric (Levenshtein distance) on $M$, which makes it a metric space. That's ...
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prove that $d$ is a metric.

Let $E=\{0,1\}^\mathbb{N}$, and $d: E\to \mathbb{R}$, defined by $d(x,x)=0$ and $$d(x,y)= 2^{-\min \left\{k\in \mathbb{N}\mid x_k \neq y_k\right\}}$$. For all $x=(x_k)_k,y=(y_k)_k \in E$, prove that $...
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Define a metric for an annulus, which makes it seem like the curved wall of a cylinder.

Can anybody please help me in understanding this question?
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Sequence of partial sums of e in Q is a Cauchy sequence.

Verify that $X_n= \{ \sum_{i=0}^n$ $\frac{1}{i!}$} is a Cauchy sequence in $Q$ with the Euclidean metric. I can't figure out how to find an $N$ that makes this work. I figure that $d(x_n,x_m) < \...
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Establish if $g_n (\alpha)=\int_a^b \ \alpha(x) \ \sin (nx) \ \cos(nx) $ converges uniformly

$$X=\{ \alpha:[a,b] \rightarrow \mathbb{R} \}$$ $\alpha''$ exists and it is continuous $$\exists \ K>0 \ : \forall \ x \in [a,b], \forall \alpha \in X: \\ \ \\ \rvert \alpha(x) \rvert, \rvert \...
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Natural embedding of Q with the Euclidean metric in R with the Euclidean metric is an isometric embedding.

The book I'm reading states this: The natural embedding of $Q$ with the Euclidean metric in $R$ with the Euclidean metric is an isometric embedding. What is the "natural embedding" of Q with the ...
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Banach fixed point theorem for a function $f_k(x) = k(x+1/x)$

Suppose $X = [1,\infty)$. The function $f_k(x) = k(x+\frac{1}{x})$ where $k\in(0,1)$ is a contraction on $X$, furthermore, $X$ is complete and $f:X\rightarrow X$. So all the requirements for the ...
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If for every $\epsilon>0$ there exist infinitely many $n$ such that $d(x_n,c)<\epsilon$ then $(x_n)$ has a subsequence converging to $c$.

Suppose $(X,d)$ is a metric space. I am trying to show that: If for every $\epsilon>0$ there exist infinitely many $n$ such that $d(x_n,c)<\epsilon$ then $(x_n)$ has a subsequence converging to ...
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Borel measurability of a subset of a product space

Let $X$ and $Y$ be compact metric spaces and let $\mathcal B_X$ and $\mathcal B_Y$ be their respective Borel $\sigma$-algebras. Let $\mu$ be a Borel probability measure on $X$ and let $\mathcal B^*...
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Pointwise convergence of Lipschitz functions from a compact space implies uniform convergence

Let $(f_n)$ be a sequence of $1$-Lipschitz functions from $(X, d_X)$ to $(Y,d_Y)$ where the first one is compact and the latter is complete (I am not sure if this matters). Let $f_n \to f$ pointwise. ...
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$(f(x_n))_{n\in \mathbb{N}}$ converges for all continuous $f$. Does $(x_n)_{n \in\mathbb{N}}$ converge?

Let $(S, d)$ be a metric space and $(x_n)_{n\in \mathbb{N}}$ a sequence in $S$. If $(f(x_n))_{n\in \mathbb{N}}$ converges for all continuous $f:S\to\mathbb{R},$ does it follow that $(x_n)_{n\in \...
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Does there exist a metric space of cardinality aleph-two that has a countable epsilon cover?

Does there exist a metric space $(M,d)$ such that $|M|=\aleph_2$ but there exists a countable $\epsilon$-cover of $M$?
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26 views

Cauchy sequences are bounded

As $\{x_n\}$ is a Cauchy sequence, there exists a positive integer $N$, such that for any $n \geq N$ and $m \geq N$, $d(x_n,x_m) \lt 1$; that is, $|x_n-x_m| \lt 1$. Put $M = |x_1| + |x_2| + |x_3| + .....
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When is an Open Set Homeomorphic to the Interior of its Closure?

Let $X$ be a topological space and $U \subseteq X$ open. Then $U \subseteq \operatorname{int}(\operatorname{cl}(U))$. I am looking for known assumptions on $X$ and $U$ such that one of the following ...
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54 views

Conformal map is an isometry

I have the upper half-plane $\mathbb H$ with the metric given by $$\mathrm ds^2=\frac{1}{y^2} (\mathrm dx^2+\mathrm dy^2)$$ and the unit disk $\mathbb D$ with the metric given by $$\mathrm ds^2=\frac{...
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Does there exist a continuous function $g:S^1 \to S^1$ such that $(g(z))^2=z , \forall z \in S^1$?

Let $S^1:=\{z \in \mathbb C:|z|=1\}$ ; does there exist a continuous function $g:S^1 \to S^1$ such that $(g(z))^2=z , \forall z \in S^1$ ?
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Compactness of infinite union under these conditions

Assume I have an infinite sequence $(S_k)_{k\in\mathbb N}$ of sets $S_k\subset \mathbb R^n$, assume that all the $S_k$ are compact with respect to the topology induced by some metric $d:\mathbb R^n\...
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Is there anything special about the below finite metric space? See below for details.

I am a high school student who has been playing around with certain mathematical ideas, most recently metric spaces, and I believe I have just "defined" if you will, the following metric space: Metric ...
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19 views

What values are used for the counter $k$ in this proof involving a convergent subsequence?

Lemma Suppose that $(x_n)$ is a Cauchy sequence in a metric space $(X,d)$, and $x\in X$. Also suppose $(x_{n_k})$ is a subsequence of $(x_n)$ such that $x_{n_k}\to x$ as $k\to \infty$. Then $x_n\to x$ ...
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Why do metric spaces that produce the same topology have different number theoretical difficulties?

Consider finding a a point with rational distance to the corners of unit square. Under the Euclidean metric this is very hard. (unsolved) Under the "city block" or taxicab metric this is very easy ...
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Does there exist a compact metric space $X$ containing countably infinitely many clopen subsets?

From this Clopen subsets of a compact metric space we know that any compact metric space $X$ contains at most countably many clopen subsets ; my question is : Does there exist a compact metric space $...
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Show that $\bigcap_{r>0}B(a,r)=\bigcap_{n=1}^\infty B(a,\frac{1}{n})=\{a\}$

Let $(M,d)$ be a metric space. Show that $\bigcap_{r>0}B(a,r)=\bigcap_{n=1}^\infty B(a,\frac{1}{n})=\{a\}$ where $B(a,r)$ is a ball with center in $a$ and radius $r$. My attempt: Set $0<r\leq ...
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Co-ordinate transformation of metric

In a past exam paper that I am using to prepare for my upcoming finals, I have encountered the following question (paraphrased): Given the metric: $$\mathrm{d}s^{2} = -c^{2}\:\mathrm{d}t^{2}+\left(...
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$X,Y$ be metric spaces , $f:X \to Y$ be a continuous and closed map , then the boundary of $f^{-1}(\{y\})$ is compact for every $y \in Y$ ?

Let $X,Y$ be metric spaces , $f:X \to Y$ be a continuous and closed map , then is it true that the boundary of $f^{-1}(\{y\})$ is compact for every $y \in Y$ ?
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Homeomorphism from $(0,1)$ to $\mathbb{R}$

I want to show that $(0,1)$ is homeomorphic to $\mathbb{R}$ by finding a homeomorphism between the two. I think the function will be related to $tan(x)$ but I'm stuck on how to modify it to fit the ...
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Subset of separable metric space can have at most a countable amount of isolated points

Let $(X,d)$ be a separable metric space. Prove that every subset $Y \subset X$ can have at most a countable amount of isolated points. Attempt at proof: Let $Y$ be an arbitrary (non-empty) subset of $...
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Topological finer and separability

I have this question: Let $(X, d_1)$ and $(X,d_2)$ be two metric spaces. Suppose $d_1$ is topologically finer than $d_2$. What is the relationship between these two statements? (i) $(X,d_1)$ is ...
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Differentiable version of Urysohn's lemma

Let $A,B$ be disjoint non-empty closed sets in $\mathbb R$ , then does there exist a differentiable function $f:\mathbb R \to [0,1]$ such that $f(A)=\{0\} , f(B)=\{1\}$ ? If the answer to the previous ...
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Given $(X,d)$ is a metric space , then the following statements are equivalent.

We need to show that if $(X,d)$ is disconnected => there exists two non-empty disjoint subsets $A$ and $B$ both open in X s.t $X= A \cup B$. I was able to prove the disjoint part , now we need to ...
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Name for function that is Lipschitz continuous over partitioning of input space

Let $f: X \to \mathbb R$ and $(X,d)$ be a metric space. Let $P=\{P_1,P_2,\dotsc\}$ be a countable partitioning of $X$. I would like to assume that $f$ is Lipschitz continuous on $(P_i,d)$ for all $P_i ...
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$X \subseteq M(n,\mathbb C)$ and $S:=\{x \in \mathbb C : x$ is an eigen value of some matrix in $X\}$ ; does $X$ closed implies/if $S$ is closed?

Let $X \subseteq M(n,\mathbb C)$ and $S:=\{x \in \mathbb C : x$ is an eigen value of some matrix in $X\}$ ; does the closed-ness of any one of $S$ or $X$ implies that the other set is also closed ?
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Open and closed sets in a $\infty$-metric space

Denote by $\mathcal{H}$ the set of continuous maps $ h:\mathbb{R}\rightarrow \mathbb{R}$ such that $h(0)=0$. We endow $\mathcal{H}$ with the supremum metric $$ \widehat{d}(f,g)=\sup\{\vert f(x)-g(x)\...
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Constructing a sequence of functions, not Cauchy

I'm working in the set $B = \{ f \in C[0,1] : \int_0^1 f(x)dx \leq 1\}$. I'm constructing an argument to show that there exists at least one sequence that has a subsequences satisfying the property ...
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$X \subseteq M(n,\mathbb C) ; |X|>1 ; $ connected/path connected, what about $S:=\{x \in \mathbb C : x $ is an eigenvalue of some matrix in $X\}$?

Let $X \subseteq M(n,\mathbb C)$ be a set with more than one element and $S:=\{x \in \mathbb C : x $ is an eigenvalue of some matrix in $X\}$. I know that if $X$ is compact then so is $S$. My question ...
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If a set is open in one metric it is open in another?

Ive been struggling to grasp a certain situation involving metric spaces and was wondering if anyone could be of any help. In the notes for my module on metric spaces I have the following "If two ...
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A discrete metric space is complete

We can read here that every discrete metric space (where the topology is the same as the discrete topology, i.e. where all the singletons are open) is complete, but an example bothers me because I don'...
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Connectedness of the sets $\{A \in M(n,\mathbb R) : \mathrm{rank}(A) =m\}$ and $\{A \in M(n,\mathbb R) : \mathrm{rank}(A)\ge r\}$

Let $r>0$ , I know that $\{A \in M(n,\mathbb R) : \mathrm{rank}(A) < r\}$ is path connected in $M(n,\mathbb R)$ . My question is ; for positive integer $m < n$ , is the set $\{A \in M(n,\...
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If $X$ is compact and $T:X \to Y$ continuous and bijective, show that $T$ is homeomorphism.

Let $X$ and $Y$ be metric spaces, $X$ compact, and $T:X \to Y$ bijective and continuous. Show that $T$ is a homeomorphism. My attempt: We need only show that $T^{-1}$ is continuous. Let $M \subset ...
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Open subsets of $\mathbb{R}^2$

Open subsets of $\mathbb{R}$ can be written as disjoint unions of open intervals- can the same be said in $\mathbb{R^2}$? Open subsets of $\mathbb{R^2}$can be written as disjoint unions of open ...
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Equality between weight and density in metric spaces

I have to prove that in any metric (in generalized version metrizable) space weight of the space is equal to its own density. My job done so far: $$(X,\delta)$$ Is topological space with metric ...
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1answer
32 views

Why the separate notation for norm

One usually denotes the norm as $\|\cdot\| $, $\| v\| := \sqrt{\langle v, v \rangle}.$ However, in metric spaces, one often writes $d(x,y) \equiv \lvert x-y \rvert$. Since the norm canonically ...
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On a function $f: \mathbb R^m \to \mathbb R^n$ , $n>1$ , mapping connected sets to connedted sets and discontinuous at a point

Let $f: \mathbb R^m \to \mathbb R^n$ be a function mapping connected sets to connected sets where $n>1$ ; let $a \in \mathbb R^m $ and $ \epsilon >0$ be such that $f(B_{\delta}(a)) \cap (\...
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Limit and Isolation points.

I have attempted a question that says to prove that the set of isolated points of a countable complete metric space X forms a dense subset of X. My Attempt It has been shown previously that the ...
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Prob. 2, Sec. 20 in Munkres' TOPOLOGY, 2nd ed: Does this metric on $\mathbb{R} \times \mathbb{R}$ induce the dictionary order topology?

Here's Prob. 2, Sec. 20 in the book Topology by James R. Munkres, 2nd edition: Show that $\mathbb{R} \times \mathbb{R}$ in the dictionary order topology is metrizable. This question has ...
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Accumulation point in real spaces

Sequences in $\mathbb{R}^n$ have a unique limit. Is it true that for any sequence which converges to limit exist there exists no accumulation point a such that $x \neq a$. i.e. does unique limit ...
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If $X$ is a non-compact metric space, can $X^n$ ever be compact?

Do there exist metric spaces $X$ such that $X^n$ is compact even though $X$ is not? Since compact spaces can have non-compact subspaces, e.g. $[0,1)\subset[0,1].$
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Showing a sequence converges weakly.

Let $f \in L_2(\mathbb{R})$. How can I show that the sequence ${g_n}$ converges weakly to $0$ in $L_2(\mathbb{R})$, where $g_n(x) = f(x − n)$? If this is not true could someone provide a counter ...
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16 views

Almost negative definite matrices and norm-distance matrices

An "almost negative definite" matrix $A$ satisfies the property $$ v^te = 0\implies v^tAv\le 0 $$ where $e=(1,1,\dots,1)$. We know that if $A$ is a simmetric zero-diagonal (hollow) matrix, then $A$ ...
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Metric Space Topology…/// [closed]

Actually I've no idea how to approach to the correct answer...
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Finite open subsets of a metric space

We know that every open subsets of a finite metric space is finite. Is it possible to have an infinite metric space (not discrete) having a non-empty finite open set? In that case the metric space ...