Metric spaces are sets on which you can measure the "distance" between any two points. The distance measurement is generally required to be symmetric (so distance from $A$ to $B$ is the same as distance from $B$ to $A$), positive for two distinct points, and obeying the triangle inequality.

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Homeomorphic and Isometric Spaces

Problem I'm currently studying metric spaces, and the lectruer's notes make the remark: Clearly $(0,1)$, $(0,\infty)$ and $\mathbb{R}$ are homeomorphic under the standard metrics, but no two of them ...
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103 views

Prove or disprove: There is a metric on $R$ with respect to which…?

Here is the text: For each point $p$ of $\Bbb R$ is $sum_p$ the family of subsets of $\Bbb R$ containing $p$ and can be obtained by deleting from $\Bbb R$ not more than countable infinity of ...
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244 views

Showing convergence in Space of Squared Summable Sequences

My Problem: Show that the sequence ${x_n}_{n\geq 1}$, where $x_n=(1,\frac{1}{2},\ldots,\frac{1}{n},0,0,\ldots)$ converges to $x=(1,\frac{1}{2},\frac{1}{3},\ldots,\frac{1}{n},\ldots)$ in $l_2$ My ...
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Cauchy-Schwarz for metrics with arbitrary signatures

When the norm of a vector is always greater than or equal to zero, the Cauchy-Schwarz inequality holds, but what if we look at a metric with an arbitrary signature? Then the inner product of a vector ...
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50 views

Property of Integrals

The Question: Assume that we are dealing with the set of all continuous functions on $[a,b]$. What can we say about $\int_{a}^b |f(x)-g(x)|dx=0$ in terms of $f(x)$ and $g(x)$. My Question: I am ...
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819 views

Do projections onto convex sets always decrease distances?

Suppose $(M, d)$ is some $\ell_p$ metric space (not necessarily Euclidean), and $C \subseteq M$ is a closed convex set. Consider the projection function $f_C:M\rightarrow C$ defined such that: ...
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87 views

Characterisation of closed balls in terms of diameters

True or false: The closed balls of a metric space are precisely those subsets such that every proper superset has strictly greater diameter.
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295 views

Metrizable group

Let $ G $ be a metrizable group. If (i) $ K $ is a closed normal subgroup of $ G $ and (ii) both $ K $ and $ G/K $ are complete, then $ G $ is complete. Here is how I am proceeding: It can be ...
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“The closure of the unit ball of $C^1[0, 1]$ in $C[0, 1]$” and its compactness

[I really want to apologize if this problem looks a little too long.] The problem : This is taken from here [Question. 3.7 (c)] and it says... Prove or disprove the comapctness of the closure of ...
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The set of points whose distance to a set $E$ in $\mathbb R^n $ is zero, is the same set $E$?

If a set $E$ is contained in $\mathbb R^n$ with the standard euclidean norm and if define another set $B$ as the points in $\mathbb R^n$ whose distance to the set $E$ is zero, is it true that $E=B$? ...
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135 views

Compact set mapped to compact set. [duplicate]

Possible Duplicate: Continuous images of compact sets are compact Theorem: Let V be a metric space. If $X\subseteq V$ and $f:X\rightarrow Y$ is continuous then $f(X)\subseteq Y$ is ...
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Tychonoff cube $I^{m}$

Different references have different definitions for the Tychonoff cube. For example; R. Engelking, General Topology: The Tychonoff cube of weight $m‎‎‎\geq‎‎‎‎\aleph‎_{o}$ is the space $I^{m}$, ...
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371 views

Help with metrics, Box topology and non-metrisable Hausdorff spaces

I'm trying to come up with a simple example of a second countable Hausdorff space that is not metrisable. The most promising I've come up with so far is the Box topology: The following is a basis for ...
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1answer
215 views

Give an example: $X, Y$ metric space, $X$ not compact, there is no $V$ for which $f^{-1}(V) \subset U$

Give an example of a non compact $X$ and sets $C$ and $U$ such that there is no $V$ satisfying the following Let $X$ and $Y$ be metric spaces, with $X$ compact, and $f: X \to Y$ continuous. ...
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$X, Y$ metric spaces, $X$ compact, $f: X \to Y$ continuous then $f^{-1}(V) \subset U$

Let $X$ and $Y$ be metric spaces, with $X$ compact, and $f: X \to Y$ continuous. Let $C$ be a closed subset of $Y$. Show that for any open neighboorhood $U$ of $f^{-1}(C)$ there is an open ...
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636 views

Minkowski's Inequality for 0<p<1

The Question: Let $X=\mathbb{C}^n$ and $0<p<1$. Define $d_p$ by $d_p(z,w)= (\sum \limits_{k=1}^{n} |z_k-w_k|^p)^\frac{1}{p}$, where $z=(z_1,z_2,\ldots, z_n)$ and $w=(w_1, w_2, \ldots, w_n)$ are ...
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265 views

Metric Spaces of Continuous Function

Let X be a metric space defined as such: $$ X = f : [0,1] \to \Re : f \,\text{ is continuous} $$ $$ d(f,g) = sup_{x\in[0,1]} | f(x) - g(x)|$$ I need to show: a) The neighborhood, $ N_r (0) $, is ...
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451 views

Metrics on $\mathbb R^{\mathbb N}$.

I was showing $D(x,y) = \sup_{k \in \mathbb N} \frac{d' (x_k,y_k)}{k}$ induces the product topology on $\mathbb R^{\mathbb N}$. Here $d(x,y)$ is the standard metric on $\mathbb R$ and $d'(x,y) = ...
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119 views

$f(\bigcap K_n)=\bigcap f(K_n)$? Where $K_n$ are compact

$f:X\rightarrow Y$ is continous map between metric spaces, $K_n$ are non empty nested sequence of compact subsets of $X$, then we need to show the title above. Please tell me which result I should ...
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3answers
203 views

Is this function necessarily a contraction?

If $(X, d)$ is a compact metric space satisfying $d(f(x), f(y)) < d(x, y)$ for all $x, y \in X$ such that $x \ne y$, is $f$ necessarily a contraction? I know an analogue of the Banach Fixed Point ...
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1answer
564 views

Standard bounded metric induces same topology

Theorem: If $(X,d)$ is a metric space and $d' = \min (d(x,y), 1)$ is the standard bounded metric then $d$ and $d'$ induce the same topology. Equivalently, for all $x_0$ there are $a,b$ such that for ...
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Example of complete metric space: $\mathbb{R}$, $d(x,y)=|e^x-e^y|$

In this thread, on the fourth question, is the following metric complete? $4. \mathbb{R}, d(x,y)=|e^x-e^y|$ The example for it is not complete was: what about $\langle -n:n\in\Bbb N\rangle$? My ...
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Open set as a countable union of spheres

Could you tell me how to prove that each open set can is a countable union of open balls: $ K\left( x,r \right) = \left\{ y \in \mathbb{R} ^{n} | \ d(x,y) <r \right\}$ where d is the Euclid ...
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Are all metric spaces topological spaces?

I think this is true but i cannot prove it. Any answer or hints are welcome. I have tried to start with $\mathbb{R}$ with euclidean metric. We may consider $\tau :=\{\emptyset,\mathbb{R}\}$ and ...
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668 views

A problem on Cauchy sequences in metric spaces.

Let $ X $ and $ Y $ be metric spaces, and let $ f: X \to Y $ be a mapping. Determine which of the following statements is/are true. a. If $ f $ is uniformly continuous, then the image of every Cauchy ...
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507 views

a multiple choice question on metric space

Let $(X, d)$ be a metric space and let $A \subset X$. For $x \in X$ define $$d(x,A) = \inf\{d(x, y) \mid y \in A\}.$$ Pick out the true statements: a. $x \mapsto d(x,A)$ is a uniformly ...
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101 views

Which of the following define a metric?

Which of the following define a metric? a. $d((x, y), (x’, y’)) = \min\{|x – x’|, |y – y’|\}$ on $\mathbb{R}^2$. b. $d((x, y), (x’, y’)) = |x| + |y| + |x’| + |y’|$ on $\mathbb{R}^2$.. c. $D((x, y), ...
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Triangle Inequality for supremum metric

Edited Heavily Here all functions are from $[0,1]$ to $\mathbb{R}$ and are bounded. Prove the following Triangle inequality in following case: ...
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a problem on Which of the following metric spaces are complete?

Which of the following metric spaces are complete? (a) The space $C^1[0, 1]$ of continuously differentiable real-valued functions on $[0, 1]$ with the metric $d(f, g) = \max_{t∈[0,1]}|f(t) − g(t)|$. ...
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Metrizability of weak convergence by the bounded Lipschitz metric

Why is the weak convergence of probability measures on $\mathbb{R}$ with respect to bounded continuous test functions $C^0_b(\mathbb{R})$ metrizable by the bounded Lipschitz metric $$d(\mu, \nu) = ...
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How to prove the metric which defined by supremum of all semi-metric?

Define the function $f:X\times X \to R$ by $d(x,y)=\sup\{d_i(x,y):i\in I\}$, when each $d_i$ is a pseudometric; $d_i(x,y)=0$ need not imply $x=y$; for every $i$ in a directed set $(I,\leq)$ and $X$ is ...
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are they complete metric spaces?

I need to know whether the following spaces are complete or not Space of all continuos real valued functions with compact support with supnorm metric The space $C^1[0,1]$ with metric ...
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Bounded subset of $\mathbb{R}$. [closed]

Let $d$ be the usual metric on $\mathbb{R}$. A subset $A$ of $(\mathbb{R},d)$ is bounded if and only if it has both a lower bound and an upper bound. Many times I have seen this paragraph as a ...
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Addition of points on Metric Space

Well, I was not quite aware that addition of points is not defined in metric spaces but is defined only on linear spaces and others. Could anyone elaborate why is this? Is the addition of intervals ...
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Finding a homeomorphism $\mathbb{R} \times S^1 \to \mathbb{R}^2 \setminus \{(0,0)\}$

Are there any specific 'tricks' or 'techniques' in finding homeomorphisms between topological or metric spaces? I'm trying to construct a homeomorphism between $\mathbb{R} \times S^1 \to \mathbb{R}^2 ...
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Proving connectedness of punctured disc without using path-connectedness

Using path-connectedness it is easy to see that the punctured disc $$D:=\lbrace(x,y)\in\mathbb{R}^2:0<x^2+y^2<1\rbrace$$ is connected. I was wondering if there is a proof that $D$ is a connected ...
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Equivalent metric generated by a compact subset of $C(X)$ where $X$ is a compact metric space.

I need help with the following problem: Let $(X,\rho)$ be a compact metric space. Prove that if $K$ is a compact subset of $C(X)=C(X,\mathbb{R})$ (i.e. continuous functions with real values) whose ...
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show that the interval of the form $[0,a)$ or $(a, 1]$ is open set in metric subspace $[0,1]$ but not open in $\mathbb R^1$

On the metric subspace $S = [0,1]$ of the Euclidean space $\mathbb R^1 $, every interval of the form $A = [0,a)$ or $(a, 1]$ where $0<a<1$ is open set in S. These sets are not open in ...
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counterexample for the fixed point theorem

give an example of a complete metric space $(X,d)$ and a mapping $T: X \rightarrow X$ which does not have a fixed point in X and satisfies; $$ d(T(x),T(y)) < d(x,y)$$ $\forall x,y \in X, x\neq y$ ...
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Totally bounded, sequentially compact, complete, bounded, closed, equicontinuous $\Rightarrow$ compact?

Related; When $K$ is compact, if $S\subset C_b(K)$ is closed,bounded and equicontinuous, then $S$ is compact? (ZF) I just edited my whole question since i think it was a bit messy. Here is my ...
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Set of Points where $X$ is locally connected is a $G_\delta$-set

I had this question on my final: Let $(X,d)$ be a metric space. Prove that the set of points where $X$ is locally connected is the countable intersection of open subsets of $X$. I wasn't quite ...
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'Every open set in $\mathbb{R}$ is the union of disjoint open intervals.' How do you prove this without indexing intervals with $\mathbb{Q}$?

In my book's exercises section I am asked to prove that every bounded, open set in $\mathbb{R}$ is the union of disjoin open intervals. Looking around the internet I have found many strategies that ...
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Pseudonormable Product Spaces

I want to prove that a product $\prod_{i\in I}X_i$ of topological vector spaces is pseudonormable only if a finite number of the factor spaces are also pseudonormable and the rest have the trivial ...
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Counterexample sought to show that squared Hellinger distance doesnt satisfy the triangle inequality

I read in a paper that the squared Hellinger distance between two densities $f$ and $g$ $$H^2(f,g)=\frac{1}{2}\int \left(\sqrt{f(x)}-\sqrt{g(x)}\right)^2 dx$$ is not a metric. I wonder if there is a ...
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metric spaces notation

Can \begin{equation} d(x,y) < 5 \end{equation} be written as \begin{equation} y \in U_5(x)\end{equation} ? I am curious because I have seen $d(x,y) < \epsilon$ be written as $ y \in ...
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92 views

showing a rational interval is disconnected

I am trying to show that the rational interval $ S= \{x \in \mathbb Q : a \leq x \leq b\} $ is disconnected in the metric space $(X,d)$ where $X=\mathbb R$ and $d$ is the standard metric ( $ ...
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179 views

Metric spaces question

\begin{equation} P = \{f \in\ C ( \mathbb{R}, \mathbb{R}) \mid f(x+2 \pi ) = f(x)\} \end{equation} be the set of $2\pi$-periodic function. 1) Show that $P$ is a subspace of $C( \mathbb{R}, ...
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478 views

Proof about diameter of a set

I could not prove the following question could you please help me? Best Regards Let $X, d(x, y)$ be a metric space. By definition, diameter of a bounded set $A ⊂ X$ is the number $diam(A)$ = ...
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143 views

Every sequence is composed of isolated points?

Let $(M,d)$ be a metric space and $\{x_n\}_{n=1}^\infty\subset M$ be a sequence. Prove that $$\forall n\in\mathbb N,\quad\exists \varepsilon> 0 \;B(x_n,\varepsilon)\cap \{x_n\}_{n=1}^\infty = ...
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In a metric space, if a set is compact, then it is closed: improving proof

Let $(M,d)$ be a metric space. If $K\subset M$ is compact, then it is closed (and bounded). Proof Let's see that $M\setminus K$ is open. Let $x\in K$ $$\exists \varepsilon_1 (x), \varepsilon_2(x) ...