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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Showing that a function is not $(d,d)-$ continuous at a point.

Let $d: \mathbb R \times \mathbb R \rightarrow \mathbb R$ be a metric: $$ d(x,y) = \begin{cases} 0 & x = y \\ |x| + |y| + 3|x-y| & x \neq y \end{cases} $$ Show that the function $f: \mathbb R ...
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Does the metric induce the topology on $X$?

Let $X=\left\{a,b,c\right\}^{\mathbb{Z}}$ and on $X$ the product topology $\tau$, where on $\left\{a,b,c\right\}$ we consider the discrete topology. On $X$, consider the metric $$ ...
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Topology in the set of matrices

Let $M_n(\mathbb{R})$ be the set of real $n\times n$ matrices. I've proved that the map $\left \|\cdot \right \| \mapsto \left \| A \right \| :=\sqrt{\text{tr}(A^tA)}$ is a norm. Then I defined the ...
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The continuity of infimum of a function

Let $(X,d)$ be a connected metric space and $(Y,d')$ is a compact metric space. Let $f$ be a continuous function from $(X\times Y,\max(d,d'))$ into $\mathbb{R}$. Because $Y$ is compact we can define: ...
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Computing the Manhattan Distance between two clusters of points. [closed]

We have two clusters of points: c1: (1, 1), (1, 2), (1, 3) c2: (2, 7), (2, 8), (2, 9) I know the Manhattan Distance formula is as follows: $d(a,b) = \sum|b_i - ...
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Componentwise Convergence in $\mathbb R^n$

I came across the following question while preparing an exercise for basic analysis: Suppose $d$ is some arbitrary metric on $\mathbb R^N$ and $(x^n) \subset \mathbb R^N$ converges to $x\in \mathbb ...
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Clarification about completeness of metric spaces

This is probably a very silly question but it bothers me for some time. We define a metric space $X$ to be complete if every Cauchy sequence in $X$ converges to some point in $X$. But any metric ...
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$\alpha$ exists so that for any points $x_n$ there is a point at average distance $\alpha$ from the $x_n$.

Let $X$ be a connected and compact metric space. Prove a real number $\alpha$ exists so that for every finite set of points $x_1,x_2,\dots, x_n\in X$ (not necessarily distinct) there exists $x\in X$ ...
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Theorem 3.7 in Baby Rudin: The subsequential limits of a sequence in a metric space form a closed set

Here's Theorem 3.7 in the book Principles of Mathematical Analysis by Walter Rudin, 3rd edition. The subsequential limits of a sequence $(p_n)$ in a metric space $X$ form a closed subset of $X$. ...
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Decide whether D is a distance function or not

Let $A$ be a set, $X:=\{x_1,...x_k\}$,$Y:=\{y_1,...,y_{k}\}$ $\subset \frak{P}$$(A)\setminus \emptyset$ subsets of the power set of $A$, both with cardinality $k$ and $d$ be a metric on ...
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$d:\mathbb N \times \mathbb N \to \mathbb R \ d(m,n)=0:m=n ; d(m,n)=1-\frac{1}{m+n}:m \neq n. $ Question convergence of sequences $(x_n)(y_n)(z_n)$

$d:\mathbb N \times \mathbb N \to \mathbb R \ d(m,n)=0:m=n ; d(m,n)=1-\frac{1}{m+n}:m \neq n. $ Question convergence of sequences $(x_n)(y_n)(z_n)$ and prove that $B_n=\{m \in \mathbb N : d(m,n)\leq ...
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Definition of interior

An interior point is defined as the following in the Euclidean space. If $S$ is a subset of a Euclidean space, then $x$ is an interior point of $S$ if there exists an open ball centered at $x$ ...
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Adding an isolated point to a Borel space

I have a Borel space $S$, which is basically a Borel subset of a Polish space. I want to add an isolated point $\alpha$ to $S$. Let $\overline{S}=S\bigcup \{\alpha\}$. Can I say that $S$ is clopen in ...
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Can the real vector space of all real sequences be normed so that it is complete ?

Let $X$ be the vector space of all real sequences . Does there exist a norm on $X$ which makes it complete ?
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What is the closure of an open ball $B_X(\mathbf{a},r)$ in $X=\mathbb{R}^n$?

Suppose we have the open ball $B_{X}(\mathbf{a},r)$ and the closed ball $\bar{B}_{X}(\mathbf{a},r)$ of radius $r$ about $\mathbf{a}\in\mathbb{R}^n=X$ with the Euclidean metric $d_2$. What is the ...
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$X,Y$ be real NLS ; $T:X \to Y$ be a linear map such that $\ker T$ is closed ; then does $T$ have closed graph?

Let $X,Y$ be real normed linear spaces and $T:X \to Y$ be a linear map with closed kernel ; then does $T$ have closed graph ? What if we assume arleast one of $X,Y$ to be complete ?
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On the dimension of a real Normed Linear Space possessing a certain property

Let $X$ be a real NLS such that for every proper subspace $Y$ of $X$ , $\exists x \in X$ such that $||x||=1$ and $dist (x,Y)=1$ ; then is $X$ finite dimensional ?
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$Y$ is a ( closed) proper subspace of a real NLS $X$ such that $dist (x,Y)=1$ for some $x \in X$ with $||x||=1$ ; is $Y$ finite dimensional?

Let $Y$ be a finite dimensional proper subspace of a real NLS $X$ , we know that we can find $x\in X$ ( depending on $Y$) , such that $||x||=1$ and $dist (x,Y):=\{||x-y||:y\in Y\}=1$ . I would like to ...
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Is this proof about metric spaces and boundaries correct?

Prove that $\partial(A\times B)=(\partial A\times \overline{B}) \cup (\overline{A} \times \partial B)$ given that $E$ and $F$ are metric spaces and contain $A$ and $B$ respectively Knowing that ...
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Is a space metric on the positive real numbers not complete?

Say we have a metric space $(\mathbb{R}^+, d)$ where the distance function is $d(x,y) = |x - y| + | 1/x - 1/y |$ Then I argue that this metric space is not complete: If we look at the Cauchy ...
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Cover $(0, +\infty )$ by open sets

Cover $(0, +\infty)$ by open sets $U_\alpha$ such that for any $\epsilon > 0$ there are points $x, y \in (0, +\infty)$ with $|x-y|<\epsilon$, not both belonging to the same $U_\alpha$ The ...
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Are $\{0\},\{1\}$ clopen in $\{0,1\}$ with the Euclidean metric? [closed]

Are $\{0\},\{1\}$ clopen in $\{0,1\}$ with the Euclidean metric? I think they are, but I would like a confirmation. Thank you.
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$X$ be finite dimensional real NLS , let $x \in X$ , does there exist $T \in \mathcal B(X)$ such that $\{T^n(x):n \in \mathbb N\}$ is dense in $X$?

Let $X$ be a finite dimensional real normed linear space , let $x \in X$ , then does there exist a continuous linear transformation $T:X \to X$ such that $\{T^n(x):n \in \mathbb N\}$ is dense in $X$ ? ...
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the point set is nowhere dense in $X$

I am trying to show for a metric space $X$, a set $\{x\}$ consisting of a single point is nowhere dense. I have proven it by showing $[{\{x\}}]^o = \emptyset$ where $[A]$ is the closure of the set $A$ ...
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Canonical metric on the suspension of a metric space

Let $(X,d)$ be a metric space. Is there a metric $d'$ on the (unreduced) suspension $\Sigma X = (X\times[-1,1])/\sim$ of $X$ such that $d'$ restricts to $d$ on $X\times \{1/2\}$? Further, we would ...
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Is it possible to prove this using set theory only and no more?

Got to prove: If $E$ and $F$ are connected sets, and $A$,$B$ are subsets of $E$ and $F$ respectively (but neither $A$ or $B$ are empty or fill $E$ and $F$entirely). Then $(A\times B)^c$ is connected ...
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Let $X$ be a non-zero real NLS , $x,y \in X$ , $B(x,r)\subseteq B(y,s)$ , then is it true that $r \le s$ ?

Let $X$ be a non-zero real NLS , $x,y \in X$ , $B(x,r)\subseteq B(y,s)$ , then is it true that $r \le s$ ? If $y=x$ then it is easy to see that that's the case . So I thought let $y \ne x$ ; I tried ...
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Fix some point $a \in M$. Prove that the function $f:M \to \mathbb R$ defined by $f(x) = d(a,x)$ is a continuous function on $M$

Let $M$ be a metric space with metric distance function $d(x, y)$, for $x, y \in M$. Assume that $M$ has only a countable or finite number of points, and assume that $M$ is connected. Fix some point ...
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Quotient of compact metric space is metrizable (when Hausdorff)?

Apparently it's a standard result that, although not every (Hausdorff) quotient of a metric space is metrizable, it always is metrizable when the space being quotiented is compact. Alas, I can't find ...
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Which of the options makes sense for this “boundary of a set” excercise

I've got a homework to do, I'm not sure if I'm missing parenthesis or something, I've got to prove this: For the metric spaces $E=(E,d_1)$ and $F=(F,d_2)$ with $A\subset E$ and $B\subset F$. ...
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Are proper maps compact?

Recently I learned about the notion of a proper map in metric spaces. Namely, if $X$, $Y$ are metric spaces, then a map $f:X\rightarrow Y$ is called proper iff for every compact set $K\subseteq Y$ the ...
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Calculate $p$-adic metric

Let $p$ be a prime. Define the $p$-adic modulus of $x$ on $\mathbb{Q}$ as $$ x= \frac{a}{b} \cdot p^{n}.$$ where $a$ and $b$ are relatively prime and do not contain $p$ as a factor as $|x|_p=p^n$. For ...
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Topology/ Metric on possibly unbounded functions

I am trying to think of a topology (possibly metric, as I am more used to think about things in metric spaces) on possibly unbounded functions (on $\mathbb{R}$) such that 1) convergence in that ...
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$d(a,X) = d(a,\overline{X})$ (distance from point to set is distance from point to closure)

I'm trying to understand this proof that: $$d(a,X) = d(a,\overline{X})$$ The proof says: Since $X\subset \overline{X}$, then $d(a,\overline{X})\leq d(a,X)$. We just need to show that the $<$ ...
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Any set of diameter $2r$ can be contained in a ball of radius $r$

So I think the following statement is correct, but I don't really know how to go about showing it: If $S \subseteq \mathbb{R}^{n}$ has diameter $2r$, can it be contained in a closed ball of radius ...
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Is there a metric that is zero for translations?

First define a relation $\sim$ on $\mathbb{Z}^k$ such that for any $a,b\in \mathbb{Z}^k$ where $a=(a_1,\dotsc,a_k)$, and $b=(b_1,\dotsc,b_k)$ we write $a\sim b$ if and only if $a-b=(n,n,\dotsc,n)$ for ...
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Numbers in $\mathbb{Q}_p$ can be written uniquely as $\sum_{i=k}^\infty \alpha_i p_i$

I'm studying the completion $\mathbb{Q}_p$ of $\mathbb{Q}$, through the classic approach of building equivalence classes of Cauchy sequences on $\mathbb{Q}$, with the $p$-adic metric. At this point ...
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Proving that $B:=\{f(x)\in C[a,b]:f(a)=0\}$ is close set

Let $B$ be a group of all the continuous functions in the interval $[a,b]$ such that $f(a)=0$. Prove that $A$ is close group in the metric space $C[a,b]$ My attempt: Metric space $C[a,b]$ ...
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Can someone suggest any way to complete (if possible) the following proof of the fact that in a not complete metric space is not compact?

Problem. If $(X,d)$ is a metric space such that it is not complete then prove that $X$ is not compact. My Attempt. Since $(X,d)$ is not complete, there exists a Cauchy sequence ...
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A map between metric spaces preserving convergent sequences is continuous

Pugh, "Mathematical Analysis", exercise 2.17: Assume $f : M \to N$ is a map from one metric space to another which satisfies the following condition: for every convergent sequence $(a_n) \subset ...
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Show that $\rho : X \times X \mapsto \mathbb{R} $ is continuous function on $(X \times X, \tau)$

Show that $\rho : X \times X \mapsto \mathbb{R} $ is continuos function on $(X \times X, \tau)$ where $\tau ((x_1,x_2), (y_1,y_2)) = \sqrt{\rho (x_1-y_1)^2 + \rho (x_2-y_2)^2}$ and $X \times X$ is the ...
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$X\subset Y \implies \overline{X}\subset \overline{Y}$ (closure inclusion subset)

Suppose that $X,Y\subset M$, being $M$ a metric space. In order to prove that: $$X\subset Y \implies \overline{X}\subset \overline{Y}$$ If $x\in \overline{X}$ we have that $d(x,X) = 0$. But I ...
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prove that $(X\cap Y)^- \subset X^- \cap Y^-$

I have to prove: $(X\cap Y)^- \subset X^- \cap Y^-$ Well, if $a\in (X\cap Y)^-$ then there is an open set $A$ containing $a$ such that: $$A\cap (X\cap Y)\neq \emptyset$$ I've tought of some ...
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Proving $(A\times B)^- = A^-\times B^-$ (closure of cartesian product)

My proof, for: $$(A\times B)^- = A^-\times B^-$$ using the metric $$d''((a_1,a_2),(b_1,b_2)) = max\{d_1(a_1,b_1),d_2(a_2,b_2)\}$$ $\rightarrow$ Well, if $a = (a_1,a_2)\in (A\times B)^-$ then: ...
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17 views

Understanding distance between point and line via infimum

The distance between point and line independently on metric is defined by $$d(X, l) = \inf\{d(X, Y)|Y\in l\}.$$ I have troubles understandning how this infimum works. Can someone please give me an ...
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1answer
44 views

Proof of $\partial\partial\partial S=\partial\partial S$

How can I prove in metric space that $$\partial\partial\partial S=\partial\partial S$$ using the proposition of the boundary below? $$\partial S=\text{cl}S \cap \text{cl}(S^c)$$ I found that if ...
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Are these metrics?

I want to find if the below functions are metrics. I have worked through each of the three conditions, but am stuck on the positivity of $f(a, b)$ (first condition-see below) and the triangle ...
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26 views

If a set is separable its completion is separable

$\tilde{M}$ is the completion of M. This question has ben done before, but with no answer. My idea is to first: show that $S$,$T$ sets (which compose the separation of M) are actually the preimage of ...
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91 views

Are $(C[0,1],d_\infty)$ and $(C[0,1],d_1)$ homeomorphic?

Two metric spaces are said to be homeomorphic if there is a bijection f between them such that $f$ and $f^{-1}$ are both continuous. Consider $C[0,1]$ with metrics: $d_\infty (f,g)=\max_{x\in ...
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23 views

Showing that $X$ is locally compact

Now I would like to change question in If $X$ is locally compact, second countable and Hausdorff, then $X^*$ is metrizable and hence $X$ is metrizable. Is it true that If $X$ is Hausdorff, second ...