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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$\epsilon$-$\delta$ continuity definition on non-compact spaces

I started studying topology and encountered the epsilon-delta definition of continuity applied for general metric spaces. From my calculus courses I am used to thinking of both $\epsilon$ and $\delta$ ...
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highway metric topologically equivalent to euclidean metric?

Consider the Euclidean metric space $(S, d_1)$ on $\mathbb{R^2}$ and the highway metric space $(S, d_h)$ on $\mathbb{R^2}$, where the highway metric is defined as $$d_h(x,y) = \begin{cases} |x_2 ...
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equivalent metric space

Let $(X, d)$ be a metric space where $d$ is unbounded, that is, $$\sup\{d(x; y) : x, y\in X\} = \infty$$ Define a bounded metric $p$ on $X$ such that: $(i).$ $f : (X, d) \rightarrow (X, p)$, $f(x) = ...
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1answer
30 views

$L_1\subset L_p$?

I am trying to check whether the implication $\forall p>1\quad f\in L_p(X,\mu)\Rightarrow f\in L_1(X,\mu)$ is true when $\mu(X)<\infty$. By $L_p(X,\mu)$ I mean the space of Lebesgue integrable ...
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28 views

Showing that $\mathcal{H}^s$ is Borel regular (assuming we know already know that $\mathcal{H}^s$ is measure)

I am trying to show that $\mathcal{H}^s$ (s-dimensional Hausdorff measure) is Borel regular. I am using the defintion $\mathcal{H}^s_{\delta}(F)=inf \Bigg\{ \sum_{i=1}^{\infty}|V_i|^s : \{V_i\} \text{ ...
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0answers
28 views

Metric space with measure and a special property

Let $R$ be a metric space endowed with a (complete) measure $\mu$ satisfying the following condition: all the open and closed sets of $R$ are measurable and for any measurable set $M\subset R$ and any ...
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2answers
36 views

Find an open set $B$ such that $g^{-1}(B)$ is not open

I cannot understand part ii) in this solution. I cannot see the significance of arbitrarily close to 0 points for which $|sin(\frac{1}{x_n})|=1$
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1answer
41 views

Aggregating Metrics to Form a New Metric

I'm looking for a source or hints which could help me solve the following problem: Let $S$ be a set and let $d_i : S \times S \rightarrow [0,1]$ be a family of metrics for $i \in \{1, \ldots n\}$. ...
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1answer
60 views

An injective continuous map on the unit sphere is a homeomorphism

Let $U$ be the set of complex numbers with magnitude $1$. Let $f: U \to U $ be an injective, continuous map. Prove that $f$ is a homeomorphism. Since $U$ is compact, it suffices to ...
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1answer
46 views

Proof with set compactness with river metric

We have got $d_r$ metric $$d_r(x,y) = \begin{cases} |x_2-y_2|, & \text{if $x_1 = y_1$;} \\ |x_2| + |y_2| + |x_1-y_1|, & \text{if $x_1 \neq y_1 $} \end{cases}$$ Prove that inside ...
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2answers
69 views

$A \subset \mathbb{R}^n$. If every continuous function $f: A \rightarrow \mathbb{R}$ is is bounded and attains its bounds then A is compact.

I'm doing a metric spaces course and got stuck on proposition. I have a feeling that I want to show that $A$ is bounded and closed then use Heine-Borel theorem. The proposition states that $f$ is ...
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1answer
45 views

Existence of a Maximal Element of the Set of Subsequential Limits of a Bounded Sequence

OK, so I've been burned by this all day now and I've given up. Supposing that we have a bounded sequence, I cannot grasp how the maximal element (as my professor put it) could exist if we have a ...
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1answer
26 views

Convergent subsequence in a bounded sequence

Let $\Phi$ be an infinite family of monotonic real functions defined on $[a,b]$ such that $$\exists C,K\geq0:\forall\varphi\in \Phi\quad(\sup_{x\in[a,b]}|\varphi(x)|\leq C\quad\land\quad ...
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0answers
64 views

Confirm solution to chapter 2, Problem 18 in Rudin's book: principals of mathematical analysis

Is there a non-empty perfect set $E$ in $\mathbb{R}^1$ which contains no rational numbers? My effort: Yes, there is. We take $E_0 \colon = [\sqrt{2},\sqrt{3}]$. Then $E_0$ is non-empty, closed, ...
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1answer
32 views

Baby Rudin Problem Chapter 2, Problems 17(c) and (d)

Let $E$ be the set of all $x \in [0,1]$ whose decimal expansion contains only the digits $4$ and $7$. Then I've managed to show that (a) $E$ is not countable, and (b) $E$ is not dense in $[0,1]$. ...
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2answers
38 views

Let A and B be disjoint closed subsets of Rn. Define d(A,B)=inf{∥a−b∥:a∈A and b∈B}. Show that if A={a} is a singleton, then d(A,B)>0.

Let $A$ and $B$ be disjoint closed subsets of $\mathbb{R}^n$. Define $d(A,B)=\inf \{||a-b||: a \in A, b \in B\}$. I have to show that if $A=\{a\}$ is a singleton set, then $d(A,B)>0$ and I have no ...
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1answer
40 views

Contraction in a complete metric space [closed]

I have this question and I need your help please. Can you please guide me how can I prove this? Thanks! $(X,d)$ is a complete metric space and $A,B\subset X$ are two closed subsets. $$\inf ...
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5answers
52 views

Let $f(z)$ be a holomorphic function on C. Show that $\overline{f(\bar{z})}$ is holomorphic on C

Since $f(z)$ is holomorphic, I used Cauchy-Riemann equations and got $u_x = v_y ,\ u_y = -v_x$ Then I wanted to check if Cauchy-Riemann equations are satisfied for $\overline{f(\bar{z})}$ It does. ...
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1answer
23 views

What is the relation between the union of the derived sets to the derived set of the union in a metric space?

Let $(X,d)$ be a metric space, and let $A$ and $B$ be two (non-empty) subsets of $X$. Let $A^\prime$, $B^\prime$, and $(A \cup B)^\prime$ denote the derived set (i.e. the set of all the limit points) ...
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1answer
61 views

Are distance functions $ d(a_1,x), …, d(a_n,x) $ for an arbitrary metric space linearly independent?

If we have a metric space $ (X,d) $, a finite set of distinct points $ a_1, ..., a_n $, do the distance functions $ d(a_1,x), ..., d(a_n,x) $ have to be linearly independent? That is, if $ ...
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Is the set of natural numbers with this metric complete?

Let $\mathbb{N}$, the set of all natural numbers, be given the metric $d$ defined as follows: $$ d(m,n) \colon= | m^{-1} - n^{-1} |$$ for all $m$, $m$ in $\mathbb{N}$. Then how to determine if ...
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2answers
253 views

Is this metric space complete?

Let $a$, $b$ be two real numbers such that $a<b$, and let $X$ be the set of all (real or complex-valued) functions defined and continuous on $[a,b]$ with the metric $d$ defined as follows: $$ ...
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2answers
51 views

Erwine Kryszeg's INTRODUCTORY FUNCTIONAL ANALYSIS WITH APPLICATIONS, Section 1.5-8

In Section 1.5-8, in his book, INTRODUCTORY FUNCTIONAL ANALYSIS WITH APPLICATIONS, Kryszeg tries to show that the set $X$ of all polynomials defined on a given closed interval $[a,b]$ on the real ...
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0answers
20 views

Finding a norm on $ \mathbb{R}^X $ such that the “natural” embedding of a metric space $ X $ in $ \mathbb{R}^X $ becomes an isometry

This question is related to this one: Every metric space can be isometrically embedded in a Banach space, so that it's a linearly independent set Kuratowski's embedding indeed seems to be the ...
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1answer
19 views

Open sets in topology and metric spaces

Let $\tau$ = { $\emptyset,[0,1],\mathbb{R}$} ($\mathbb{R},\tau$) is a topological space, right? Since the intersection of any of the sets in $\tau$ is itself in $\tau$, and same for the union. But ...
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1answer
69 views

Integral equation and metric spaces

Let $C([0,\frac{\pi }{2}])$ be the set off all continuous functions defined on $[0,\frac{\pi }{2}]$ . Prove that this integral equation $$ f(t) = \int\limits_0^{\frac{\pi }{2}} {\arctan } ...
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k-Cells are Connected

I am studying real analysis from Baby Rudin, and while the book proves that real intervals are connected, it does not say anything regarding k-cells. I would expect them to also be connected, but do ...
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1answer
80 views

Metrics on X. Show that they are equivalent if and only if…

Suppose that $d$ and $ρ$ are metrics on a set $X$. Prove the following statement: The metrics $d$ and $ρ$ are equivalent if and only if the class of $d$-open sets of $X$ exactly coincides with the ...
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1answer
35 views

Show that the metric $d_1$ on $C([0,1])$ does not give rise to a complete metric space. [duplicate]

Show that the metric $d_1$ on $C([0,1])$ does not give rise to a complete metric space. $$ d_1(f,g) = \int_0^1 |f(s)−g(s)| \, ds $$
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metric characterization for connectedness

Is there a metric characterization of connectedness? I'm looking for something like the following metric characterization of compactness: A metrizable topological space is compact if, and only if, ...
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1answer
51 views

Endowing an abelian group with a metric.

I solved the following exercise, which is not hard: Let $G$ be an additive abelian group, such that exists $f: G \to \mathbb{R}$ satisfying: $f(0) = 0$ and $f(x) > 0$ for all $x \neq ...
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1answer
11 views

Connected spaces of $M(n,\mathbb R)$

Consider $M(n,\mathbb R)$, the space of all $ n\times n$ matrices over $\mathbb R$.Which of the following are connected? a.$O(n)$ the set of all orthogonal matrices b.$GL(n,\mathbb R)$ set of all ...
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How is $\sqrt{2}$, for example, in the closure of $\mathbb{Q}$ in the usual metric space $\mathbb{R}$?

Let $\mathbb{R}$ be the set of all real numbers under the usual metric $d$ defined as follows: $$d(x,y) \colon= |x-y|$$ for all $x$, $y$ in $\mathbb{R}$, and let $\mathbb{Q}$ be the set of all ...
2
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1answer
37 views

Compact subsets of $M(n,\mathbb R)$

Consider $M(n,\mathbb R)$, the space of all $ n\times n$ matrices over $\mathbb R$.Which of the following are compact? a.$O(n)$ the set of all orthogonal matrices b.$GL(n,\mathbb R)$ set of all ...
0
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1answer
26 views

Are there properties of vector space equipped with two norms?

I am interested in a vector space equipped with two norms$ \lvert \lvert \cdot \rvert \rvert$ and $ \lvert \lvert \cdot \rvert \rvert ^*$ satisfies that there is $M>0$ such that $ \lvert \lvert x ...
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2answers
22 views

Equivalance of norms

Let $X$ be the vector space of all real valued functions defined on $[0,1]$ having continuous first-order derivatives. How to show that the following norms are equivalent: $\|f\|_1 = |f(0)| + ...
4
votes
1answer
62 views

How to decide completeness of $\ell^\infty$?

Let $\ell^\infty$ denote the set of all bounded sequences $x \colon = (\xi_j)_{j=1}^\infty$, $y \colon= (\eta_j)_{j=1}^\infty$ of complex numbers with the metric $d$ defined as follows: $$ d(x,y) ...
0
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1answer
40 views

Calculate distance in x,y from center based on distance and degrees.

I'm terribly sorry if this question is written like a 5-year old.. But that's the level I'm on in terms of math and coordinate calculations. (Just realized I don't even know what to tag this question ...
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2answers
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path metrics without geodesics

This is a follow-up of this question. Recall that a metric space $(X,d)$ is called a path-metric space if the distance between any two points in $X$ equals the infimum of lengths of paths between ...
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Is the following function uniformly continuous?

I am supposed to prove if the function $ f:X \to \mathbb {R}, f (x) = dist (x, A)$ where $ A$ is an arbitrary subset o the metric space $X $ is uniformly continuous. If both points $ x $ and $ y $ ...
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1answer
51 views

Every metric space can be isometrically embedded in a Banach space, so that it's a linearly independent set

I'm trying to prove the following: Every metric space can be isometrically embedded in a Banach space, so that it's a linearly independent set. I came up with the following idea: Let $ (X,d) $ be a ...
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1answer
26 views

Prove Contraction Mapping

The following is given: Eucliden metric $d$, defining the distance between vector $v_1=(x_1,y_1)$ and $v_2=(x_2,y_2)$: $d(v_1,v_2)=\sqrt{(x_1-x_2)^2 + (y_1-y_2)^2}$ $M $ is a mapping of $\mathbb ...
0
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1answer
28 views

$C([a,b] \times [c,d],X)$ compared to $C([a,b],C([c,d],X))$ and $C([c,d],C([a,b],X))$

Let $C(Y,X)$ be the space of continuous functions from $Y$ to $X$ together with the supremums norm. Here $Y$ is a compact space and $X$ a metric space. Let $a,b,c,d \in \mathbb R$ be finite, with ...
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1answer
29 views

Metric Spaces, Continuity and Preference Relations

Let X be a metric space and $\succeq$ be a preference relation on X. The preference relation is continuous if the sets $\succeq (y) =\{x: x \succeq y\}$ and $\preceq (y) = \{x : x \preceq y\}$ are ...
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1answer
26 views

How to prove continuity of addition over weird metric? Edit: Ignore this. Errors in the problem definition.

Let $f: R \times R \rightarrow R$ and let the metric over $R$ be $d(x,y)=|x-y|$ and let the metric in $R \times R$ be $d_2((x,y),(a,b))= ((x-y)^2+(a-b)^2)^{1/2}$. I believe I understand how to ...
0
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1answer
26 views

Finding open balls in $\mathbb{R}^2$

If anyone can help I would be highly grateful! The Problem is in the image below.. [1] http://i.stack.imgur.com/eKDH2.png Should you approach using the open balls to find the boundary of the set? ...
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1answer
16 views

Drawing Ball (0,1) in half Euclidean Metric [closed]

Just a quick question. I know it will be a circle of radius 2, but can somebody just clarify why?
3
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1answer
76 views

shortest path in complete metric space

Let $(X,d)$ be a complete connected by arcs metric space. We define the length of a continuous path $\gamma: [0,1] \rightarrow X$ to be \begin{equation*} \sup\limits_{0=a_{0}<a_{1}<... a_{n}=1} ...
0
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1answer
20 views

When are the following inclusions $\subsetneq$

When does the "equality" part of inclusion fail in: $$\overline{A \cap B} \subseteq \overline{A} \cap \overline{B}$$ and $$Int(A \cup B) \supseteq Int(A) \cup Int(B)$$ ? Can you provide an simple ...
0
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1answer
22 views

continuity of a metric d

from Continuity of the Metric and Convergence Sequences, why $d^{-1}(V)$ is an open ball? to be an open ball, I think it contains elements of $X$, not $X^{2}$. why is it?