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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complete subset of a metric space

Let $f:X\to Y$ be a continuous map between metric spaces. Then $f(X)$ is a complete subset of $Y$ if A. the space $X$ is compact B. the space $Y$ is compact C. the space $X$ is complete D. the ...
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Is a Banach space $X$ Lipschitz equivalent to the metric quotient $X/B$, where $B$ is the closed unit ball?

Recall that the metric quotient $X/B$ is defined as follows: first we consider the equivalence relation $\sim$ on $X$ that identifies all points of $B$, then we define on the set of all equivalence ...
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Prove that this series converges?

I have a Banach space $X$ and a linear operator $A \in L(X)$. $A$ is bounded such that $||A|| <1$. I then have to show that $$log(I-A)=\sum_{n \ge 1} \frac {A^n}n$$ converges. All I can come up ...
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Proving completeness and compactness of a sequence of metric spaces.

The problem statement Let $(X_n,d_n)_{n \in \mathbb N}$ be a sequence of metric spaces. Consider the product space $X=\prod_{n \in \mathbb N} X_n$ with the distance $d((x_n),(y_n))=\sum_{n \in ...
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Completeness of metric spaces

edit: Another question about formatting. My questions on math.stackexchange keep cutting off the last few lines of my post. How do I fix this? The remaining lines show up in my edit box but not in the ...
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Proving two statements about locally compact spaces

The problem statement: Let $(X,d)$ be a locally compact metric space (for every $x \in X$, there exists a compact neighbourhood of $x$) $a)$ Prove that if $K_1 \subset X$ is compact, then, there are ...
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Hypersphere isometry?

I will denote the $n-$sphere of radius $1$ centered at the origin as $\mathbb{S}^n$, so that $$ \mathbb{S}^n = \{ x \in \mathbb{R}^{n+1}\ : \ \|x\| = 1\}. $$ I am stuck on the following problem...I'm ...
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Prove that the space of divergent sequences in $(l_{\infty},d_{\infty})$ is open and dense. Is it separable?

The problem statements are: Consider the space $A=\{ \{a_n\}_{n \in \mathbb N} \in l_{\infty} : \{a_n\}_{n \in \mathbb N} \text { is not convergent }\}$ $a)$ Prove that $A$ is open and dense in ...
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51 views

Prove this sequence in $L(\ell_1)$ does not converge to zero?

I have a linear operator $$A: \ell_1 \rightarrow \ell_1$$ $$A_nx=(0,0,...,0,x_{n+1},x_{n+2},...)$$ with $n$ zeros. I am asked to show two things: that for any $x \in \ell_1$, $\lim_{n \rightarrow ...
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Proving a continuous function $f:K \cup A \to \mathbb R$ is uniformly continuous if $K$ is compact and $A$ is discrete.

Let $(X,d)$ be a metric space. Let $K \subset X$ compact and $A \subset X$: $\exists \delta>0$ such that $d(a,b)>\delta$ for all $a,b \in A$ with $a \neq b$. Consider in $K \cup A$ the induced ...
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Symmetrical endomorphisms and quadratic forms

(This last part of my linear algebra course is causing me quite a bit of headaches, so please be patient) Let $V$ be a vector space over the real field, and we'll indicate with $(\cdot,\cdot)$ its ...
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Topologically equivalent metrics but not strongly equivalent in$Lip_{M}(\mathbb{R})=\{f: [0,1] \rightarrow \mathbb{R} : |f(y)-f(x)|\leq M.|y-x| \}$

Let's consider this set $Lip_{M}(\mathbb{R})=\{f: [0,1] \rightarrow \mathbb{R} : |f(y)-f(x)|\leq M.|y-x| \}$ (i.e Lipschitz functions in $[0,1]$). How can I prove that $(Lip_{M}(\mathbb{R}), ...
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Is a set $U$ consisting of the single point $p$ open or closed?

I'm guessing here that $U$ would have to be closed, especially since for example the proof of the theorem that the union of two closed sets is closed is also valid if one of the sets is $U$. Still, ...
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Proving the set of “distance functions” on a compact set is a compact set itself

The problem statement. Let $(X,d)$ be a compact metric space and $C(X)=\{\phi: X \to \mathbb R : \phi \text{ is continuous}\}$. For each $x \in X$ we define the function $f_x: X \to \mathbb R$ ...
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Prove or disprove two statements about open functions on metric spaces

Let $f: (X,d) \to (Y,d')$ an open function (not necessarily continuous) between metric spaces. Decide whether the following statements are true or false: 1) If $A \subset X$ doesn't have isolated ...
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Uniform implies pointwise convergence

I had a question to show a sequence of functions $(x_n)$ in $C[0,1]$ (equipped with a metric $d$) does not contain a uniformly convergent subsequence. $$ x_n(t) = \ n(1-nt) \ \ \ \ \ \ \forall \ ...
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Show that A=$\{(x_1,…x_n) \in \Bbb R | -1\le x_1\le x_2\le …x_n\le 1\} \subset \Bbb R^n $ is closed.

The full question was: Show that A=$\{(x_1,...x_n) \in \Bbb R | -1\le x_1\le x_2\le ...x_n\le 1\} \subset \Bbb R^n $ is compact, but I was able to show correctly that it is bounded. However my ...
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How to show the intersection of two compact subsets is compact

Let (X,d) be a metric space and A,B $\subset$ X be two compact subsets. Show $A\cap B$ is also compact. I attempted this question by showing the intersection is bounded and closed. But I stated ...
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329 views

Deciding whether two metrics are topologically equivalent in the space $C^1([0,1])$

Consider the space $C^1([0,1])$ and the function $d:C([0,1])\times C([0,1]) \to \mathbb R$ defined as $d(f,g)=|f(0)-g(0)|+sup_{x \in [0,1]}|f'(x)-g'(x)|$. Decide whether the metrics $d$ and ...
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139 views

Proving a subset $A$ of a metric space $(X,d)$ is open

Let $(X,d)$ be a metric space and let $A \subset (X,d)$. Prove that $A$ is open iff for every sequence $\{a_n\}_{n \in \mathbb N}$ such that $lim_{n \to \infty} a_n \in A$, there exists $n_0 : a_n \in ...
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basic question involving topology and the Hausdorff distance

Let $\Omega_1 \supset \Omega_2 \supset ...$ a sequence of nonempty , open, bounded and convex sets. Define $\Omega = int \Bigl( \overline{\displaystyle\bigcap_{k=1}^{\infty} \Omega_k } \ \Bigl) $ and ...
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Prove this is a bounded linear operator and find its operator norm?

I have a map $$A:(C[0,1], || \centerdot ||_\infty) \rightarrow \mathbb R, Ax = x(0) \forall x \in C[0,1]$$ and need to prove it's a bounded linear operator, and find its operator norm. I've tried ...
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On local rings $(R,m)$ having a metrizable $m$-adic topology

If $R$ is a local ring with maximal ideal $m$ and the intersection of powers of $m$ is $0$, then the $m$-adic topology is metrizable. Is there a condition on $R$ assuring that the metric space so ...
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71 views

Proving that a metric on space of analytic functions is equivalent to compact convergence

Let $U\subseteq \mathbb C$ be open and $\mathscr A(U)$ consist of all analytic functions on $U$. I can easily prove that there exists a sequence $K_n$ of compact sets in $U$ so that ...
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Continuous bijection whose inverse is not continuous at uncountably many points

I am interested in understanding to what extent continuous bijections fail to be homeomorphisms. For example, suppose $X, Y$ are metric spaces and $f: X\to Y$ is a continuous bijection. Is it possible ...
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Has the idea of generalizing the codomain of a metric been seriously considered?

The long line is much longer than $\mathbb{R}$, and indeed many chains have this property. Thus, since metrics are usually assumed to be real-valued, this can be understood as an assumption that ...
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$\mathbb Q$-metric spaces: how much is lost?

I want to know how weak (literally "punctured") the theory of metric spaces becomes if we impose the condition that the distance function can assume only rational (nonnegative) values.
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On eventually constant sequences

It is of course true that in a discrete space a sequence converges iff it's eventually constant. Is the converse true, i.e., if the only convergent sequences in a space are eventually constant, is the ...
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54 views

Demonstrate that the following metric space is not compact

Let $X$ be a metric space. Show, if there is an $r > 0$ and a sequence $(x_n)$ from $X$ such that $d(x_n,x_m) \geqslant r$ for $n≠m$, then $X$ is not compact. I know that sequentially compact and ...
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Metrizable topological space $X$ with every admissible metric complete then $X$ is compact

How to prove: If $X$ is a metrizable topological space and every admissible metric on $X$ is complete then $X$ is compact. I was trying with an idea of contradiction and thereby to construct ...
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730 views

Finding the “differentness” of two point clouds

I would like to reduce the "differentness" of two point clouds $X$ and $Y$ to a single comparable value $\lambda$, which would ideally be $0$ when $X$ and $Y$ are identical upto isometry (rotation, ...
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'Compactness' vs 'Closed and bounded' for general metric spaces

We know that for $A \subset \mathbb{R}^n$, A is closed & bounded $\iff$ A is compact and that this does not generalize to general metric spaces. 1.) For which class of metric spaces, is ...
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If every subset of X is open then there are no limit points in X

Let $(X,d)$ be a metric space. Then if every subset in $X$ is open there are no limit points in $X$. This is probably a very simple question, but I just can't seem to get anywhere with it.
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Prove that Baire space $\omega^\omega$ is completely metrizable?

When I tried to prove that Baire space $\omega^\omega$ is completely metrizable, I defined a metric $d$ on $\omega^\omega$ as: If $g,h \in \omega^\omega$ then let $d(g,h)=1/(n+1)$ where $n$ is the ...
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Maurice Frechet's 1904 Definitions of Compactness

I'm writing a small paper on the history of compactness. Frechet wrote in French, and I don't speak French, so I've been consulting this paper: Taylor, A.E. On page 244, I read that Frechet proved ...
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find examples to prove $ A \cup B $ is not part of the $ V $

For $ A, \: B $ is the subspace of $ V $ Find examples to prove $ A \cup B $ not a subspace of $ V $ I learn about this program should not know how. Desire to help people and give a solution ...
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is symmetric chi-squared distance “A” metric?

Is symmetric chi squared distance $$\int \frac{(p-q)^2}{pq}\mbox{d}\mu(x)$$ a metric? I am searching web since long time ago but I couldnt find anything. It is positive and is zero whenever $p=q$ ...
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Countable product of complete metric spaces [duplicate]

Someone that can give me a proof of that a countable product of complete metric spaces is complete ?
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Selecting a subset from a set such that a given quantity is minimized

Let $A$ be is a set of some $p$-dimensional points $x \in \mathbb{R}^p$. Let $d_x^A$ denote the mean Euclidean distance from the point $x$ to its $k$ nearest points in $A$ (others than $x$). Let $C ...
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About the continuity of a function in the closed graph theorem proof

I'm reading Functional Analysis book of Rudin, and in the proof of the closed graph theorem, there's one point that I don't understand. Can someone please explain it to me? I really appreciate this. ...
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Each open cover of a sequentially compact metric space has Lebesgue number

I want to query, whether I'm right. (I'm sorry if don't use the correct words in my translation, please feel free to correct, and give me hints.) I have a metric space $(X,d)$ which is sequentially ...
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Application of fixed point theorem in $R^n$

Let $A=(a_{ij}) \in \mathbb R^{n \times n}$ a matrix such that $|a_{ij}|<\frac{1}{n}$ for every $i,j$. Prove that $I-A$ is invertible. My attempt at a solution: $I-A$ is invertible $\iff$ ...
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Functions on finite metric spaces

Is a function f from a finite metric space M to itself always continuous? I have tried proving it, but I have gotten stuck. Any help would be appreciated.
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$(X,d)$ m.e., with $Y \subset X$: $Y$ is open, $Y$ is connected it's equivalent to another property.

Let $(X,d)$ be a metric space and let $Y \subset X$: $Y$ is open. Prove that $Y$ is connected if and only if there aren't $A,B \subset X$ non-empty such that $Y=A \cup B$ and $A \cap \overline ...
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Composition of two bijective constant displacement maps

Let $(M,d)$ be a metric space. A constant displacement map is a function $f$ from $M$ to $M$ such that $d(x,f(x))=d(y,f(y))$. My question is this: Is the composition of two bijective constant ...
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How to show the metric space is complete?

the space is the Real line with bounded metric (i.e. $d/(1+d)$, $d$: euclidean). We thought that since nd the space real line with euclidean metric is complete and the bounded metric is smaller than ...
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$A = \left\{ (1,x) \in \mathbb{R}^2 : x \in [2,4] \right\} \subseteq \mathbb{R}^2$ is bounded and closed but not compact?

Is it true that set $A = \left\{ (1,x) \in \mathbb{R}^2 : x \in [2,4] \right\} \subseteq \mathbb{R}^2$ is bounded and closed but is not compact. We consider space $(\mathbb{R}^2, d_C)$ where ...
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Metric-like families of relations

Let $X$ be an arbitrary set and to start with, let us consider a relation $\leq$ on $X$ (that is $\leq$ is a subset of $X^2$) which is reflexive and transitive. such a relation is called a preorder. ...
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80 views

Partition and open cover in compact metric space

I'm having troubles solving this problem, any help will be appreciate :) Let $X$ be a compact metric space and for all open cover $U$ of $X$ we denote $N(U)=min\{|V| : V $subcover of $ U\}$. Let ...
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116 views

How to show the space is totally bounded?

the space the Real line with bounded metric (i.e. d/(1+d), d: euclidean) we know that totally boundedness means that there exists a finite epsilon-net. we first approached to question by directly try ...