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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Prove that if $f$ is continuous on a compact set then it is uniformly continuous

Prove that if $f$ is continuous on a compact set then it is uniformly continuous. Proof: Let $f:A\rightarrow \mathbb{R}$ be a continuous function and let $A$ be a compact subset in a metric ...
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Density In The Theorem/Proof of The Stone-Weierstrass Theorem

Yet again, I have a question that I could use some help with. Note that almost everything can be found in C. Pugh's, Real Mathematical Analysis (soft-cover, 2nd Edition, ISBN: 978-1-4419-2941-9); ...
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Let $U\subset \mathbb R^n$ and $Q=\{(x_1,…,x_n)\in U\mid x_1=1\}$. Show that $Q$ is not open and $\partial Q=\emptyset$.

Let $U\subset \mathbb R^n$ and $Q=\{(x_1,...,x_n)\in U\mid x_1=1\}$. Show that $Q$ is not open and $\partial Q=\emptyset$. To me it looks obvious, but how can I write a proof properly ? This is my ...
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An open ball in $C[0, + \infty)$

Consider the space $C[0, +\infty)$ of all continuous, real-valued functions on $[0, + \infty)$ with metric $$ d( \omega_1, \omega_2 ) = \sum_{n=1}^{\infty} \frac{1}{2^n} \max_{t \in [0,n]} ( \min ...
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Infinite metric space has open set $U$ which is infinite and its complement is infinite

Let $(X,d)$ be a metric space where $X$ is an infinite set. Prove that the space has an open set $U$ such that both $U$ and its complement are infinite sets. I have considered if $d$ is the ...
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A symmetric function

While working on a research problem on fuzzy metric spaces, I came across a special symmetric function $F_n:X^n\times (0,\infty)\to [0,1]$ i.e. \begin{equation*} ...
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In a complete metric space with no isolated points , show that the intersection of open and dense sets with a countable set is non-empty.

Let $(X,d)$ be a complete metric space with no isolated points. I want to prove that for every $(Gn)$ sequence of open and dense subsets of $X$ and for every countable set $A$$\subseteq$$X$ we have ...
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53 views

Sum of Closed Subsets of $\mathbb{R}$ [duplicate]

If $A,B\subset \mathbb{R}$ are closed in $\mathbb{R}$, is $A+B$ also closed in $\mathbb{R}$? I think it is not, but could not find a counter example: any suggestions?
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Hausdorff dimension calculation of union of sets

$F$ is a Cantor set in $(0,1)$, $\dim_HF=1/5$. What's the $\dim_HE$ where $E=(F×R)\cup(R×F)$? By the product properties, I know that and $\dim_H(F×[0,1])=6/5=1+1/5$, which is the sum of hausdorff ...
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Is squared Euclidean distance a metric? [duplicate]

Is squared Euclidean distance a metric? In particular does it obey triangle inequality? I think no, but cannot find a counterexample. Edit: Does this (not obeying the triangle rule) happen only when ...
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The complement of a first category set in X is a set of second category.

Let X be a complete metric space. Then the complement of a first category set in X is a set of second category in X. What is explain in my class is "if the complement of a first category set is a set ...
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Suppose $f : X \to Y$ is a (continuous) bounded map.Does this implies that $f$ is uniformly continuous?

It's well known that if $ f : \bf (X,d) \to \bf (Y,e) $ is a uniformly continuous function then $f$ maps bounded set to bounded set.Does the converse hold ? More Precisely, Suppose $f : X \to Y$ ...
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Prove that two closed and disjoint sets, have open disjoint super-sets with dist metric .

For starters let's define the distance between a point $x$ and a set $A\subseteq X$ of a metric space $(X,d)$ as follows: $$\text{dist}(x,A)=\inf\{d(x,a):a \in A\}.$$ Now let's assume that $A,B$ are ...
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Why the word 'space' is used with metric space?

I am new to math. According to the Wikipedia, A metric space is a set for which distances between all members of the set are defined. I have a silly question: Why they used the word 'space'? Why not ...
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$S(f) = \underset{x \in X}{\sup } f(x)\ $ is continuous.

I'd like to prove that the function $ S: \mathcal{B} (M;\mathbb{R}) \rightarrow \mathbb{R}$, $S(f) = \underset{x \in X}{\sup } f(x)\ $ is continuous. $\mathcal{B}(M;\mathbb{R})$ is the set of all ...
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Where can I learn more about “approximate isometries”?

Let $Y$ and $X$ denote metric spaces and $f : Y \leftarrow X$ denote a function. Definition 0. Call $f$ an approximate isometry iff for all $x \in X,$ we have that for all $\varepsilon \in ...
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adapting proof to show interval connected

I've read and understand the proof I've attached here proving [a,b] is connected. The notes then say that is can easily be adapted to open, half open, and unbounded intervals. How would I adapt it for ...
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Metric that satisfies every property but symmetry

Suppose that we have a function $d:M\times M\to \mathbb{R}$ which satisfies every property of a metric except we have that $d(x,y)\neq d(y,x)$. Is there any interesting theories that arise from this ...
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Is the Cartesian product of finitely many metric spaces also a metric? If so, what about completeness?

Let $n$ be a positive integer, and let $p$ be a real number such that $p \geq 1$. Let $(X_1, d_1), \ldots, (X_n, d_n)$ be metric spaces, and let the set $X$ be given by $$X \colon= \Pi_{k=1}^n = X_1 ...
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a weak notion of flow in a metric space

I am seeing the definition of flow in a metric space : $f:M\times \mathbb{R}\rightarrow M$ is one flow if $M$ is metric space, $f$ is continuous and $f(x,t+s)=f(f(x,t),s)$ Note that the condition is ...
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Hausdorff dimension of F and f(F)

We have F being a subset of R, [-1,1], while f:R->R, where f(x)=x^2. What's the Hausdorff dimension of F and f(F)? I think the dim(F)=2(length) and dim(f(F))=1, is it correct? Thanks,
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Why does this proof fail?

I'm reading some notes on topology, and the notes' author is trying to raise motivation to consider compactness by providing a theorem whose proof is built intentionally wrong, but I don't agree with ...
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Does every non-singleton connected metric space $X$ contains a connected subset (with more than one point) which is not homeomorphic with $X$?

Does every non-singleton connected metric space $X$ contains a connected subset (with more than one point) which is not homeomorphic with $X$ ? Also ; does every connected metric space $X$ contains ...
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Proving a function to be a contraction

I'm studying contractions and I am trying to understand under which conditions a function is actually a contraction. I have understood that a continuous function $f: [a,b]\rightarrow\mathbb{R}$ is a ...
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Space of Lipschitz Functions Complete?

Consider the subspace of continuous, real-valued functions on $[0,1]$ that are Lipschitz. Is this subspace complete under the sup norm ($\Vert \cdot \Vert_{\infty} = \sup \{ |f(x)| : x\in S \}$)? I ...
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Find a cauchy sequence that converges outside of space

Let $h$ be the space of infinite sequences $x=(x_1,x_2,...,x_n,...)$ with a finite number of elements that are not zero. The metric defined on $h$ is $d(x,y) = \sup_{n \in \mathbb N}|x_n-y_n|$. ...
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Continuity of $v: \mathcal{B} (M;N) \times M \rightarrow N$, $v(f,x) = f(x)$.

Let $(M,d_M)$, $(N,d_N)$ be metric spaces and $v: \mathcal{B} (M;N) \times M \rightarrow N$, $v(f,x) = f(x)$. Then $v$ is continuous at $(f_0,x_0)\in \mathcal{B}(M;N) \times M$ $\iff$ $f_0 : M ...
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Taxicab metric and Hausdorff space

Consider (ℝ,Ƭ) with the topology induced by the taxicab metric. Using the definition for Hausdorff, give an example of why (ℝ,Ƭ) is Hausdorff. The finite complement topology on ℝ is not Hausdorff. ...
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Are these conditions also sufficient for a metric to be induced by a norm?

Let $(X,d)$ be a metric space such that the set $X$ is also a vector space over the field $\mathbb{R}$ of real numbers or the field $\mathbb{C}$ of complex numbers. Then the following holds: If ...
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Hausdorff distance and isometry

Let $(M,d)$ be a metric space and $\Phi (M) $ be the collection of all bounded and closed subsets $X \subset M$. For all $X,Y \in \Phi(M)$, we define their Hausdorff distance by: $$ \rho (X,Y) = ...
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Finding a bijection between $c$ (set of real sequences that converge) and $c_0$ (set of real sequences that converge to $0$)

Actually I have to prove that there is an uniform homeomorphism between $c$ and $c_0$ (regarding the $sup$ metric), but I can't even find a bijection between them in the first place. I tried to ...
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One of Hermitian metric's properties?

We now define a Hermitian manifold is a complex manifold in which unmixed components of metric tensor vanish $g_{ij}=g_{\bar{i}\bar{j}}=0$. Is this a propert of a Hermitian manifold? Or is it an extra ...
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$X$ compact Hausdorff with $X=X_1\cup X_2$. If $X_1,X_2$ are closed and metrizable, show that $X$ is metrizable.

This is Exercise 9 from Section 34 of Munkres - Topology. Following the hint given, I've done the following:Since $X$ is compact, $X_1,X_2$ are compact metrizable and hence have countable bases. Let ...
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Rotates in metric L_n

Let's define distance between the points in 2-d space: $d = \sqrt[n]{(x_1 - x_2) ^ n + (y_1 - y_2) ^ n}, n > 2$. The rotates in this space seem confusing to me. How to define the angle between ...
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Why is the intersection of an open set and a single point open?

I'm reading baby Rudin as independent study and he has the following theorem on page 36 (second edition): Let $X$ be a metric space. Suppose $Y\subset X$. A subset $E$ of $Y$ is open relative to $Y$ ...
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characterization of normed space(L2)

if $\left\|\frac{f_n(x)}{x-i}\right\|_{2}=\displaystyle\left(\int_{\mathbb{R}}\left|\frac{f_n(x)}{x-i}\right|^2d\mu(x)\right)^\frac{1}{2} \rightarrow 0$ as $n \rightarrow \infty$ such that $f_n(x) \in ...
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Regularity of Borel measures on compact metric spaces

Let $(X,d)$ be a compact metric space and $\mu$ be a finite measure on the Borel $\sigma$-algebra $B_X$ on $X$. Then we have for all $A \in B_X$: $\mu (A) = \inf \{\mu(Q) \ \vert \ A\subseteq Q, Q \ ...
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Given $a,b \in S^n$, then there exists an isometry $f: S^n \rightarrow S^n$ such that $f(a) = b$

$S^n = \{x\in \mathbb{R}^{n+1} : \|x\| = 1\}.$ I am using this definition: isometry is a surjective function $f:M \rightarrow N$ between two metric spaces $(M,d)$ and $(N,\rho)$ such that $$\rho ...
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Prove that $[a,b]$ is compact

Let $a<b$ be real numbers. Prove that $[a,b]$ is compact. Below I present my solution. I thought it's good enough, but my TA said it's incorrect. I don't see where there is a problem. Could ...
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Is $\{ (x,y) \mid -1\leq y\leq 1 \}$ open or closed?

Let $$B:= \{ (x,y) \mid -1\leq y\leq 1 \}.$$ Is it correct to say that $B$ is closed, because $$B^C := \{ (x,y) \mid \lvert y\rvert >1 \}$$ is open?
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Compact Metric Spaces and Separability of $C(X,\mathbb{R})$

Let $(X,d)$ be a compact metric space. Show that $C(X,\mathbb{R})$ is a separable metric space (space of continuous functions from $X$ to $\mathbb{R}$). I first showed that if $(X,d)$ is compact, ...
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Let $(M,d)$ be a metric space and $X \subset M$ a discrete subset.

Then, for each $x \in X$, there exists $B_x = B(x;r_x)$ such that $x \neq y \Rightarrow B_x\cap B_y = \emptyset$. My attempt For each $x\in X$, let $r_x = \inf\{ d(x,a): a\in X-\{x\}\}$. If we ...
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Connected Unbounded Subset of a Normed Vector Space

Let $(X, ||.||)$ be a normed vector space and let $V \subset X$ be a connected unbounded set for which $0 \in V$. Show that $\forall \epsilon > 0$, then $\exists x \in X$ such that ...
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Let $E$ be a normed space and $b\in E$. Then $d(b, \overline{X}) = d(b,X)$.

$d$ is the distance between a point and a set: $d(b,X) = \underset{x\in X}{\inf}\{\|b-x\| \} $ and $X = B(a;r) = \{\|a-x\|<r: x \in E\}$, $ \overline{X} = B[a;r] = \{\|a-x\| \leq r: x \in E\}$ ...
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Compact Metric Spaces which are Groups and Topologies

Let $(G,d)$ be a compact metric space which as well is a group. Assume that $(x,y) \mapsto xy$ is continuous as a map $G \times G \to G$ and that group inversion $x \mapsto x^{-1}$ is continuous as a ...
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A proof that the closure of a set A is equal to the union of A and its limit points

Wanted to check if this proof is valid (not completely sure about the end): We claim that $$\bar A = A \cup A'$$ Proof: Take some a $\in A \cup A'.$ If $a \in A$ then $a \in \bar A$. If $a \in A'$ ...
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Prove that Prove $\left\lvert \left\lVert x \right\rVert - \left\lVert y \right\rVert \right\rvert \le \left\lVert x-y \right\rVert$

This is Exercise 3.1.4 from Economic Dynamics, Theory and Computation by John Stachursky. Key definitions for the exercise are his definition of norm and metric I believe. Let Prove $\left\lVert ...
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Proving that a closed plane in $\mathbb R^3$ has a point in the plane closest to an arbitrary point outside the plane

Let $A= \{\ x \in \mathbb R^3: |x_1| + 2|x_2| +|x_3|^3 = 1\ \}$ and let $P \in \mathbb R^3\setminus A$. Show that there exists a point $y \in A$ that is closest to $P$ among all points in $A$. Assume ...
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Product of metric spaces

Let $(M,d)$ and $(N,p)$ be metric spaces. Consider the space $M \times N$ endowed with the metric $D=((x,y),(x',y'))=\max\{d(x,x'),p(y,y')\}$ for $(x,y),(x',y') \in M \times N$. Let $A ...
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Which of the following is true for a metric space? [duplicate]

Let $(X,d)$ be a metric space. Which of the following is possible? $X$ has exactly 3 dense subsets. $X$ has exactly 4 dense subsets. $X$ has exactly 5 dense subsets. $X$ has exactly 6 dense ...