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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To show that the function $f(x) = \inf \{d(x,x_n) : n \in \Bbb N \}$ is uniformly continuous on X.

Let $(x_n)$ be a sequence in a metric space $(X,d)$. Show that the function $f(x) = \inf \{d(x,x_n) : n \in \Bbb N \}$ is uniformly continuous on X. I have started the proof in this way... Let ...
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4answers
158 views

Non-empty finite point set is closed

Subset of $\Bbb R^2$: My book says that non-empty finite point sets are closed. Why is this? Since it is a finite point set, it necessarily has no limit points within it, since every neighborhood of ...
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1answer
272 views

Is $C(\mathbb R)$ Separable?

I'm working on an exercise from Carother's chapter11 of Real Analysis that talking about Space of Continuous Functions: Here, $C(\Bbb R)$ is the set of continuous real-valued functions on $\Bbb R$, ...
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1answer
107 views

Is $C(\mathbb R)$ Complete?

I'm trying to prove an exercise from Carthers' book chapter10 of Real Analysis, problem claimed as, where $C(\mathbb R)$ denote the infinity norm space of all continuous functions on real line. I ...
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1answer
31 views

Numerical approximation to the Wasserstein metric?

Are there numerical methods for approximating/calculating the Wasserstein metric in particular cases? Suppose that $f$ and $g$ are two density functions with the same support. How can I calculate the ...
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1answer
48 views

Is set $\{xy=1\}$ connected set in $4$-dim complex plane

Is a set $\{xy=1\}$ is connected in $4$-dimensional complex plane (over real number set).
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57 views

Metric under which $C([0,\infty );\mathbb{R})$ is a Polish space.

Does there exist a metric such that $C([0,\infty ); \mathbb{R})$ is a separable complete metric space? The usual supremum norm isn't even a metric on this space and I've tried several variants e.g. ...
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2answers
60 views

Why does $f(U)$ is open for every open $U\subset M$ not imply $f$ is continuous?

Let $f:M \to N$ be a map from a metric space $M$ to a metric space $N$. Does "$f(U)$ is open for every open $U\subset M$" imply $f$ is continuous? I think it's wrong but I can't find a counter ...
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If $X$ is complete, then there is no continuous and open $\,f:X \to \mathbb{Q}$

I've encountered the following question and got stuck : There is no continuous and open mapping $\,f:X \to \mathbb{Q},$ where $X$ is a complete metric space. I thought it had something to do with ...
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1answer
57 views

$\mathbb{R}$ -trees are CAT(0) space

An $\mathbb{R}$-trees is a metric space $(X,d)$ such that there is a unique geodesic segment (denoted $[x,y]$ ) joining each pair of points $x,y\in X$ ; if $[x,y]\cap[y,z]=\{y\}$ , then ...
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1answer
446 views

Prove existence of disjoint open sets containing disjoint closed sets in a topology induced by a metric.

Question: Let $(X, d)$ be a metric space. Let $A$ and $B$ be disjoint subsets of $X$ that are closed in the topology induced by $d$. Prove that there exist disjoint open sets $U$ and $V$ such that ...
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1answer
24 views

Proof that larger open balls contain the closure of smaller ones

Easy question, but I can't think of the correct way to do it. If I have a sequence of open balls, $\{U_m\}$, the balls of radius $\frac{1}{m}$ about zero, how do I show that $\overline{U_m}$ is ...
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0answers
83 views

Proof for Norms in Vector Spaces

Prove that if a norm $\|x\|$ on a real vector space satisfies the parallelogram law, then the polarization identity defines an inner product and that the norm associated with this inner product is the ...
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0answers
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Proving the Urysohn's metrization theorem by using the Nagata-Smirnov's metrization theorem

I need to prove the Urysohn's metrization theorem by using the Nagata-Smirnov's metrization theorem. Urysohn's metrization theorem: Every regular second-countable topological space is ...
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1answer
124 views

Proving that a metric is non-negative

I wanted to try a problem, where I need to prove that the non-negativity of a metric follows from the following axioms: For a metric $d$ in some space $X$, we have for $x,y,z\in X$ ...
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Distance $D(A, B)$ is defined to be $D(A, B) = \inf d(a, b)$. Show that $D$ does not define a metric on the power set of $X$ [closed]

(Distance between sets) The distance $D(A, B)$ between two nonempty subsets $A$ and $B$ of a metric space $(X, d)$ is defined to be $D(A, B) = \inf d(a, b)$ where $a$ is an element of $A$, $b$ is an ...
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1answer
43 views

$x \in \mathbb R^d$, $B$(closed)$ \subseteq \mathbb R^d$. Where $\mathbb R^d$ is an inner product space. Proof help [duplicate]

$x \in \mathbb R^d$, $B$(closed)$ \subseteq \mathbb R^d$. Where $\mathbb R^d$ is an inner product space. Show that there exists a point $b_0 \in B$ such that $d(x,B) = \|x - b_0\|$ My attempt: ...
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1answer
62 views

Closed spaces in lp and distances

I try solve exercise from Diestel: If Y is a proper closed linear subspace of $l_p$ (1 $<$ p < $\infty$), then there is an x $\in$ $S_x$ so that distance $d(x, Y)$ $>=1$. I try solve this ...
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1answer
64 views

show that for any x,y element of X, |D(x,B)-D(y,B)| < d(x,y)

The distance D(x, B) from a point x to a non-empty subset B of (X, d) is defined to be D(x, B)= inf d(x, b), where b is an element of B. So I must show that for any x,y is an element of X that ...
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55 views

a problem about compactness and sequential compactness in metric space

Consider a metric space $(\Bbb N, d)$ where $d(m,n) = \frac{\vert m-n \vert} {1+\vert m-n \vert}$. Need to prove that any infinite subset $X \subset \Bbb Z$ is not compact and not sequentially ...
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1answer
118 views

Topology of the ring of formal power series

I'm interested in defining a topology on the ring $R[[X_i]]$ of formal power series in $(X_i)_{i\in I}$, where $R$ is a topological ring and $I$ is a (possibly infinite) index set. The wiki article ...
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Closure, interior, and boundary of a subset

Consider $\Bbb R^2$ with standard distance and a subset $A= \left([0,1] \times [1, \infty ) \right) \cup \bigcup _{n=2}^ \infty \{(x, \frac1n):0 \le x \le \frac1n \}$ Question: Is A closed? Find its ...
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1answer
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If f is uniformly continuous in a metric space, there is limit in set closure

$ f:X \subset \mathbb{R}^m \to \mathbb{R}^n $ uniformly continuous in $ X $, show that $ \forall a \in \overline{X} $ exists $ \lim_{x \to a} f(x) $. Is enough to say: Since f is uniformly ...
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1answer
53 views

Continuty of functions inside a open ball

Let $ f: X \subset \mathbb{R}^p \to \mathbb{R}^q $ and $ a \in X$. Supose that for all $ \epsilon > 0 $ exists $ g: X \to \mathbb{R}^q $ continuous at $a$ such as $ \| f(x) - g(x) \| < \epsilon ...
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1answer
76 views

Computation of the extrinsic curvature tensor for a warp drive metric.

In Miguel Alcubierre's renowned paper discussing a "warp drive" metric, he discusses the extrinsic curvature. Here is an extract. My questions are quite trivial to someone who understands the ...
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84 views

Closure, interior and boundary of $(0, 1)$ with Zariski topology

If we consider $\mathbb{R}$ together with the Zariski topology, what is the closure, interior and boundary of $(0,1)$? A set is closed iff it is either finite or $\mathbb{R}$ under this topology, so ...
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3answers
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Compact Sets Metric Spaces

Lets $(\Bbb R,|x-y|)$ be a metric space. By the Heine-Borel theorem, it obviously follows that $\Bbb Q$ is not a compact set. Now, if I were to consider $\Bbb Q \cap[-1,0]\subset\Bbb R$ is that a ...
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Proof that the composition of two contractions on the same metric space (X,d) is also a contraction

I am required to prove that given the metric space $(X,d)$, a contraction $T : X \to X$ and another contraction $S : X \to X$, the compositions $T \circ S$ and $S \circ T$ are also contractions. I ...
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1answer
23 views

To show $\{(x,y,z) : x+z^2\sin(x+y) \ge z \}$ is closed in $\mathbb R^3$ by elementary methods

How to show that the set $\{(x,y,z) : x+z^2\sin(x+y) \ge z \}$ , where $x,y,z$ each are from the set of real numbers , is a closed subset of $\mathbb R^3$ under usual Euclidean metric ? I know how to ...
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1answer
38 views

Function continuity outside a closed subset

Let $f:M \subset \mathbb{R}^p \to \mathbb{R}^q $,continuous at $a \in M $. Show that if $f(a) \notin \overline{B} (b,r) \subset \mathbb{R}^q $, then exists $ \delta > 0 $ such as $ f(x) \notin ...
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34 views

Show that $d_V(x,y)$ is metric

Question: On the set of integers $\mathbb{Z}$, show that the function $d$, defined as follows, is a metric: $$d_V(x,y) = \begin{cases} 0 & \text{if}\ x=y \\ \min{\{\dfrac{1}{n!}}\}\mid\ n!\ ...
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1answer
132 views

Equality on functions in $ \mathbb{R}^n $

Let $ f,g : M \subset \mathbb{R}^p \to \mathbb{R}^q $ continuous. Given $ a \in M $, supose that all open ball centered in $a$ contains a point $x$ such as $f(x) = g(x) $. Show that $ f(a) = g(a) $. ...
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1answer
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Separability of the Space of all Real-Valued functions over $[a,b]$ with a Continuous First Derivative

I'm reading Neal Carothers' Real Analysis and I'm stuck on the following question: Let $f$ be real-valued, continuously differentiable function over $[a,b]$ and let $\epsilon>0$. Show that there is ...
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42 views

Both $F$ and $C$ are closed sets but their sum $F+C$ is not closed. [duplicate]

In context to the question what will be an counter example such that both $F$ and $C$ are closed sets in $ \Bbb R^n$ but their sum $F+C$ is not closed in $ \Bbb R^n$?
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1answer
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Spaces vs. Structures

Examples of spaces I've come across include vector spaces, inner-product spaces, and metric spaces. Examples of structures I've met include rings, fields, and groups. I have always understood spaces ...
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Exponents for Hölder functions on metric spaces

Sometimes people talk about Hölder functions on metric spaces without mentioning the allowed range for the exponent. On manifolds, it's traditional to assume an exponent in $[0, 1]$, since all ...
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1answer
42 views

Metric spaces - Limit points and Isolated points

I apologise for this, but I have numerous potential misunderstandings. $\Bbb R^2$ is a metric space, since it is a subspace of the Euclidean space, $\Bbb R^n$ I can look at a set $E$ with elements ...
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3answers
59 views

E is closed if every limit point of E is a point of E?

E is closed if every limit point of E is a point of E? Should that be "E is closed if every point of E, is a limit point"? I don't understand. Limit points are essentially points that hug other ...
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Understand the definition of convex metric spaces

I am trying to understand the following definition: We call a set $E\subset \Bbb R^k$ convex if>$$\lambda x+(1-\lambda)y\in E$$ Whenever $x\in E, y\in E$ and $0\lt \lambda \lt 1$ Clearly ...
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Understanding part of a theorem of Calculus of Variations

I have trouble understanding the following statement (From Gelfland's Calculus of Variations book): If $\phi[h]$ is a linear functional and if ...
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Density character of a metric subspaces

Is it true that if $M$ is a metric space and $N$ is a metric subspace of $M$ (I mean, $N\subseteq M$ and the metric defined on $N$ is the same metric on $M$ restricted to $N$) then the density ...
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2answers
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Every $p$-norm ($p \in [0,\infty]$) generates the same class of open sets on $\mathbb{R}^n$

The following claim has been made in my multivariable analysis class, and I think I have the idea of the proof but I can't quite seem to get down to the rigorous proof the instructor wants: Every ...
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1answer
61 views

Why isn't the completion of $C^0$ wrt. the $L^2$ norm a space of sequences instead of a space of functions?

We know that $L^2(\Omega)$ can be defined as the completion of $C^0(\Omega)$ with respect to the norm $$\left(\int_\Omega |u|^2\right)^{\frac 12}.$$ But strictly speaking, $L^2(\Omega)$ is a space of ...
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To show that $X = (0,1]$ is complete .

Show that $X = (0,1]$ is complete with respect to the metric $e $ where $e(x,y) = |\frac{1}{x} - \frac{1}{y}|$. My proof: let $(x_n)$ be Cauchy in $(X,e)$. Let $(t_n) := \frac{1}{(x_n)}$. Then ...
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Does topological equivalence of metrics imply strong equivalence?

I know that if $(X,d_1)$ and $(X,d_2)$ are metric spaces and for some positive constants $a,b$ , $ad_1(x,y) \le d_2(x,y) \le b d_2(x,y) $ for every $x,y$ in $X$ , then a subset $A$ of $X$ is $d_1$ ...
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1answer
46 views

How to prove this statement?

I cannot prove this proposition directly . Let $(X,d)$ and $(Y,d')$ be metrice spaces. Let $f$ be a function from $X$ to $Y$. If $\overline{f^{-1} ( B)} \subseteq f^{-1}( \overline B)$ for all ...
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3answers
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To show that $d $ and $ e$ are equivalent.

On the set $X = (0,1]$, consider the usual metric $d(x,y) = |x-y|, (x,y \in X) $ and another function $e: X\times X \to R$ given by $e(x,y) = |\frac{1}{x} - \frac{1}{y}|$. Show that $d $ and $ e$ ...
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3answers
132 views

Prove the intersection of a compact set and a set with no accumulation points is finite

Let $S\subset\mathbb{C}$. We say that $z_0$ is an accumulation point of $S$ if for every $r>0$, the intersection $D(z_0,r)\cap S$ is an infinite set. Let $U\subset\mathbb{C}$ be an open set such ...
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1answer
58 views

Open set in Hilbert Cube.

Any open set in the Hilbert Cube is the union of open subsets of the form $$U_1 \times ... \times U_n \times X_{n+1} \times .... \times X_{n+k} \times...$$ where $X_k := [0, \frac{1}{k}]$ for $k \in ...
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1answer
35 views

Let $A$ be any subset of $\mathbb R^{+}$ , then there exist a metric space $(X,d)$ such that $d:X \times X \to A \cup \{0\}$ is a surjection?

Let $A$ be any subset of the set of positive real numbers $\mathbb{R}_+$ ; then does there exist a metric space $(X,d)$ such that $d\colon X \times X \to A\cup\{0\}$ is a surjection ?