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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show: $\overline{\overline X} = \overline X$

is my proof correct? Definition: Let $X\subset\mathbb R$ and let $x'\in\mathbb R$, we say that $x'$ is an adherent point of $X$ iff $\forall\epsilon>0\exists x\in X \text{ s.t. }d(x′,x)≤ε$. the ...
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31 views

Showing compactness of complete metric space

I need to show that for $K>0$, $$X=\{f:[0,1]\rightarrow [0,1]\mid |f(x)-f(y)|\leq K|x-y|\ \forall x,y \in [0,1]\}$$ with the metric $d(f,g)=\max|f(x)-g(x)|$ , (supremum metric), is a compact ...
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46 views

$d(x_n,y_n)$ converges to a limit when $x_n, y_n$ are Cauchy sequences

Let $(X,d)$ be a metric space and $x_n, y_n$ Cauchy sequences. Is there a way to prove that $\lim\limits_{n \to \infty} d(x_n,y_n)$ exists without involving the completion of $X$? Intuitively you ...
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46 views

Proof that the middle-thirds Cantor set has no isolated points

Let $x_0$ be some point in the Cantor set $C$. Prove that $\forall\epsilon>0\, \exists y\in C$ such that $y\neq x_0$ and $|x_0 - y|<\epsilon$.
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62 views

Not Quite Metrization

Let's say I have a space $X$ with a function $d\colon X \times X \to \mathbb R$ that has the following 2 properties: $d(x,y)\ge 0$ for all $x$, $y \in X$ and $d(x,x) = 0$ for all $x$, $y \in X$. ...
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0answers
17 views

“Limit set” of infinite measure for a “Cauchy” sequence

Let $\{A_n\}$ be a sequence of sets $A_n\subset X$ of finite Lebesgue measure $\mu$ with the property that$$\forall\varepsilon>0\quad\exists N\in\mathbb{N}^+:\forall n,m\geq N\quad\mu(A_n\triangle ...
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2answers
58 views

Give an example of a set that is closed but not compact nor bounded. Prove your answer.

Let $X = (0,\infty)$ with the usual topology in $\mathbb{R}$ and the the usual metric. Consider $A \subset X$ where $A = [1, \infty)$. Then $A$ is closed as $A' = (0,1) \subset X$. My attempt is as ...
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1answer
79 views

Every closed set in a separable metric space is the union of a perfect set and a set which is at most countable

Prove that every closed set in a separable metric space is the union of a (possibly empty) perfect set and a set which is at most countable. (Rudin's Principles of Mathematical Analysis, 3rd ...
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22 views

Show that there exists $\epsilon >0$ such that $\bigcup_{x\in A}B(x;\epsilon)\subset V.$

Let $X$ be a compact metric space, $A$ a closed subset of $X$ and $V$ an open subset of $X$. Suppose $A\subset V$. Show that there exists $\epsilon >0$ such that $$\bigcup_{x\in ...
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39 views

Does $d(x,y) = \lvert N(x) - N(y)\rvert$ satisfy the triangular inequality?

Let $N(x)$ be the norm of the vector $X$ and efine $$d(x,y) = |N(x) - N(y)|$$ I want to prove that $d(x,y)$ satisfies the triangular inequality. Here is my attempt: $$|N(x) - N(y)| \leq |N(x)| + ...
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1answer
29 views

Closedness of Continuous Mappings from Compact Metric Space to Compact Metric Space

Let $(X, \rho_{X})$ and $(Y, \rho_{Y})$ be two compact metric spaces. Consider the metric space $(M_{XY}, \rho)$, where $M_{XY}$ is the set of any mappings from X to Y and $\rho(f,g) := \sup_{x \in ...
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23 views

Continuous functions dense in $L_p(X,\mu)$ if $X$ has a special property

Let $X$ be a metric space endowed with a measure $\mu$ satisfying the following condition: all the open and closed sets of $X$ are measurable and for any measurable set $M\subset X$ and any ...
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1answer
52 views

Approximation of $f\in L_p$ with simple function $f_n\in L_p$

Let us use the definition of Lebesgue integral on $X,\mu(X)<\infty$ as the limit$$\int_X fd\mu:=\lim_{n\to\infty}\int_Xf_nd\mu=\lim_{n\to\infty}\sum_{k=1}^\infty y_{n,k}\mu(A_{n,k})$$where ...
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37 views

$\epsilon-\delta$ continuity definition domain

Does epsilon-delta continuity implicitly requires that there would be at least one non-trivial Cauchy sequence converging in the function's domain? Generally the criteria is introduced with no ...
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2answers
38 views

prove: a complete metric space $X$ is compact if and only if …

Let $X$ be a complete metric space. Suppose that for any infinite subset $A$ of $X$ and for any $\epsilon>0$ there are $x_1,x_2 \in A$ such that $d(x_1,x_2)< \epsilon$. Show that $X$ is ...
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27 views

Convergence in $L_p$ and elsewhere

Let $\|f\|_p:=(\int_X|f|^pd\mu)^{1/p}$ and let $L_p$ be the space of (the classes of equivalence of) complex or real measurable functions such that $\int_X|f|^p d\mu<\infty$ exists. In ...
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1answer
70 views

Show that ${\mathscr C}(\{1,..,n\},R)$ and $R^n$ have the same open sets

Question: Let X be the set $\{1,2,...,n\}$ equipped with the discrete metric ($\delta(x,y)=0$ if $x=y$, $\delta(x,y)=1$ if $x\neq y$). Then ${\mathscr C}(X, R)$ and $R^n$, where $R$ is the real ...
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3answers
77 views

Domain for $\epsilon-\delta$ continuity definition

Does epsilon-delta continuity implicitly requires that there would be at least one non-trivial Cauchy sequence converging in the function's domain? Generally the criteria is introduced with no ...
2
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1answer
48 views

$\epsilon$-$\delta$ 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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35 views

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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38 views

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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27 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
27 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
35 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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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
54 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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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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61 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
42 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
24 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
61 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, ...
2
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1answer
30 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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1answer
34 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
48 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
21 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
58 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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48 views

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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244 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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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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18 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
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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
63 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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3answers
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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
77 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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123 views

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, ...
3
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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
10 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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106 views

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 ...