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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Metric spaces Lipschtiz mapping proof

Prove that the map $f : R^2 → R$ , $f(x, y) = 2 \sin x − y$ is a Lipschitz mapping with Lipschitz-constant $2\sqrt{2}$. You can use the fact that $\sqrt2\sqrt{a^2 + b^2} ≥ |a| + |b|$ So if f(x,y) ...
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An example where $f:X \to X$ is not a contraction map but $f \circ f$ is?

Can anyone give me one example where $X$ is a complete metric space, $f:X \to X$ is not a contraction map, but $f \circ f$ is? I thought in terms of having a unique fixed point, also but couldn't ...
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Which of these spaces is metrizable?

The question: Which of the following topological spaces are metrizable? Let $X$ be any non-empty set, and let the topology consist only of the empty set $\emptyset$ and the full space $X$. Let $X$ ...
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$A$ is open as a subset of $Y$ $\Leftrightarrow$ it is the intersection with $Y$ of a set which is open in $X$

The problem: Let $Y$ be a subspace of a metric space $X$, and let $A$ be a subset of the metric space $Y$. Show that $A$ is open as a subset of $Y$ $\Leftrightarrow$ it is the intersection with $Y$ ...
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Is $\tau_{d_1}$ equal to $\tau_{d_\infty}$?

$X=C[0,1]$ $f,g\in X$ , $d_{\infty}(f,g)=\max|f(x)-g(x)|,\mbox{ for}\quad 0\le x\le 1$ $d_1(f,g)=\int^1_0|f(x)-g(x)|dx$ since $d_1(f,g)\le\int^1_0d_{\infty}(f,g)dx$ $d_1(f,g)\le ...
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Question about completeness of the bounded functions as a metric space.

$(a)$ Let $S$ be a non-empty set (finite or infinite), and consider the space $X = \mathscr{B}(S)$ of all bounded, real-valued functions on $S$. Define $d_{\infty}$ on S by: $d_{\infty}(f, g) = ...
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Show that if $C(K)$ is separable, then $K$ is metrisable, for $K$ compact and Hausdorff

My question is simply as the title states: Let $(K,\tau)$ be a compact Hausdorff (topological) space. Show that if $C(K)$ is separable, then $K$ is metrisable. Firstly, I appreciate that this is ...
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Use of a Covering theorem

where I cannot see how the highlighted equation has been obtained. I cannot understand how the setminus operation has been justified. The books gives no justification.
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is $GL(n,\mathbb R)$ dense in $M(n,\mathbb R)$

Is $GL(n,\mathbb R)$ dense in $M(n,\mathbb R)$? I have proved it to be open,not closed,not connected but not sure about this property .How to do this?
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How to Prove a Metric Space is Sequential? [duplicate]

Let's say I have a space X with a "d-metric" on X, a function d:X×X→R that has the following 2 properties: d(x,y)≥0 for all x, y∈X and d(x,x)=0 for all x, y∈X. The d-ball of radius r centered at x ...
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Equivalent definition about bounded set in metric space

I have read some book they said that "A nonempty subset $A$ of metric space $X$ is bounded if $\sup \{ d(x,y) : x, y \in A \} < \infty$" and another book they said that "A nonempty subset $A$ of ...
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Is the set of all $n\times n$ matrices with determinant $1$ an open subset of $M(n,\mathbb R)$?

Is the set of all $n\times n$ matrices with determinant $1$ an open ,dense connected subset of $M(n,\mathbb R) $ i.e set of all matrices over $\mathbb R$? I know it will be a closed subset of ...
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d-metrizable spaces are sequential

By a d-metric on a set $X$ we mean a function $d : X × X \to \mathbb{R}$ satisfying the following two properties: $d(x,y)≥0$ for all $x, y∈X $ and $d(x,x)=0$ for all $x, y∈X.$ The $d$-ball of ...
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Given two different metrics on the same set, define a third different from the other two.

Given two different metrics defined on the same set, define a third different from the other other two. Prove that the third metric is, indeed, a metric. EDIT: Apologies, posted it before I could ...
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Let X, Y be topological spaces and let y ∈ Y . Show that the map i : X → X × Y, i(x) = (x, y) is continuous

I have a feeling the solution is to do with the pre image of an open set in XxY being open in X, but I'm not sure how to go about proving it.
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Baby Rudin Problem 2.29

Here's is Prob. 29 in the Exercises following Chap. 2 in PRINCIPLES OF MATHEMATICAL ANALYSIS by Walter Rudin, 3rd edition: Prove that every open set in $\mathbb{R}^1$ is the union of an at most ...
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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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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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$d(x_n,y_n)$ 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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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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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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“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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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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Every closed set in a separable metric space is the union of a perfect set and a set which is at most countable [duplicate]

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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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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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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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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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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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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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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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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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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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 ...
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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: $(i).$ $f : (X, d) \rightarrow (X, p)$, $f(x) = ...
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$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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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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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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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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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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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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$A \subset \mathbb{R}^n$. 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
75 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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67 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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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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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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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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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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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) ...