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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Show completeness of metric subspace

I have problems solving the following 2 problems: Given is the metric $d:\Bbb R\times\Bbb R\to[0,\infty[$ with $$d(x,y):=|\arctan(x)-\arctan(y)|\;.$$ a) Show that the metric subspace ...
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25 views

Questions about proof of $\lim x_n = a, \lim y_n = b\implies \lim x_n+y_n = a+b$ in a normed vector space

I need to prove that, in a normed vector space $E$, we have: $$\lim x_n = a, \lim y_n = b\implies \lim (x_n+y_n) = a+b$$ and: $$\lim\lambda_n = \lambda, \lim x_n = a \implies \lim \lambda_n\cdot ...
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Example of an uncountable metric space where every point is isolated

I was trying to come up with an example of an uncountable metric space all of whose points are isolated. I've had difficulty thinking of one, has anyone got any nice examples? Just in case: ...
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Cantor's Intersection Theorem Exam Problem

I hate submitting questions on Stack Exchange without providing an attempt at answering the question, but I genuinely don't know how to answer the following exam problem. I've asked my professor to ...
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18 views

$\Sigma_2^+$ is complete given metric $d$

Let $\Sigma_2^+$ be the set of functions $f : \mathbb{N} \to \{1,2\}$. I wish to show that this set endowed with the metric $$ d(s,t) = \sum_{j = 0}^\infty \frac{\vert s_j - t_j \vert}{3^j} $$ is a ...
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27 views

Show that $\{f_n(x) \}_{n \in \mathbb{N}}$ doesnt converge in M.

Let $n \in \mathbb{N}, f_n(x)=e^{-(x-n)^2}$ and $g(x)=\Bigg\{\begin{array}{ll} 1 & x=0 \\ \frac{1-e^{-x^2}}{x^2} & x \neq 0 \\ \end{array} $ You can assume that g is continuous ...
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29 views

If two sequences are Cauchy, then d(sequence_1, sequence_2) is cauchy in R

The question says this: If $(X,d)$ is a metric space and $\{x_n\}$ and $\{y_n\}$ are Cauchy sequences, prove that $\{d(x_n,y_n)\}$ is a Cauchy sequence in $R$. I see that I would have to show that ...
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1answer
25 views

If $X$ is totally bounded then every sequence contains a Cauchy subsequence

I attempted the proof, I just want to see if it is correct: Suppose $X$ is totally bounded and $(x_n)$ is a sequence in $X$. Then $(x_n)$ has a subsequence contained in a ball of radius $1/2$. This ...
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1answer
22 views

Show that [0, 1) with the induced topology from R is a Polish space.

It's easy to see that the space is separable because $Q \cap [a,b)$ is a countably dense subset of $[a,b)$, but I can't figure out a way to show that it's completely metrizable. I know this means ...
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1answer
68 views

Showing a metric space is not complete.

Consider the metric space $$B = \{ f \in C[0,1] : \int_a^b \left| f(x) \right| dx \leq 1\},$$ where $d(f,g) = \int_0^1 \left| f(x) - g(x) \right|dx$. I'm trying to show that this metric space is not ...
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36 views

Questions about Proof that Cartesian Product of Open Sets is an Open Subset

I'm trying to understand the proof that: The cartesian product $A_1\times \cdots\times A_n$ of open subsets $A_i\subset M_i$ is an open subset of $M=M_1\times\cdots\times M_n$. It follows ...
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2answers
44 views

Does the Hausdorff property hold on closed subsets of $\mathbb{R}^n?$

I am trying to prove that given disjoint closed $A,B\subseteq \mathbb{R}^n$, there exist disjoint open $U,V$ containing $A,B$ respectively. In other words that we can take the Hausdorff property to ...
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40 views

homeomorphism from interval $[a,b]$ to $[0,1]\subset \mathbb{R}$

I need to show that every interval $[a,b]$ is homeomorph to $[0,1]\subset \mathbb{R}$. I've found this answer but it only deals with open sets, and I need an answer that deals with closed sets.
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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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32 views

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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1answer
17 views

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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19 views

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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1answer
40 views

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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28 views

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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1answer
17 views

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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1answer
109 views

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 ...
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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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1answer
35 views

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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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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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 ...
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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 ...
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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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1answer
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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 ...
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1answer
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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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1answer
37 views

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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1answer
28 views

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} = ...
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1answer
70 views

$X,Y$ 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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1answer
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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 ...
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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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1answer
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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 ( I am also considering $M(n,\mathbb R) \subseteq M(n,\mathbb C)$ and $S:=\{x \in \mathbb C : x $ is an eigenvalue of some matrix ...
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3answers
48 views

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 ...
2
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1answer
28 views

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 ...
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1answer
42 views

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 ...
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1answer
38 views

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 ...