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.

learn more… | top users | synonyms (1)

-4
votes
1answer
41 views

Intersection of metric spaces [closed]

Do you think that the intersection of two metric spaces (X, $\mathcal T$) and (X, $\mathcal T'$) is a metrizable space, or at least is a Hausdorff space ? If this is not the case, would you have any ...
0
votes
2answers
24 views

Continuity in Metric Spaces between two spaces under a function f

Let (X,d) and (Y,e) be metric spaces , and let f:X→Y be a function. Explain but do not prove if the statement is correct. If there exists r>0 so that $e((f(x1),f(x2))$$≤$ $r(d(x1,x2))$for every ...
1
vote
1answer
31 views

Projection map is open

Let $X=X_1$ x $X_2$ where $(X_1,d_1)$ and $(X_2,d_2)$ are metric spaces. Equip $X$ with a product metric $d$. Define a map $\Pi_1:X \to X_1$ by $\Pi_1(x_1,x_2) = x_1$. Let $U \subset X$ be open and ...
0
votes
1answer
22 views

Continuously differentiable functions dense in $L^2[a,b]$

I read in Kolmgorov-Fomin's Элементы теории функций и функционального анализа (p. 408 here) that the set of continuously differentiable functions are dense everywhere in space $L^1[a,b]$ of Lebesgue ...
0
votes
2answers
16 views

Subsets of uniformly discrete sets are closed

Let $(X,d)$ be a metric space. Suppose that $A$ is a countably infinite subset with the property that there exists some $\varepsilon>0$ such that if $a,b\in A$ and $a\neq b$, then ...
7
votes
4answers
231 views

Does a continuous point-wise limit imply uniform convergence?

Question Given a sequence of continuous functions $(f_n)_{n \in \mathbb N}$ and define $$ f : X \rightarrow Y, \quad f(x) = \lim_{n \rightarrow \infty} f_n(x) $$ where $X$ and $Y$ are metric spaces. ...
1
vote
2answers
44 views

Covering of Complete Metric Spaces

Baire's Theorem says that if $X$ is a complete metric space and $$X=\bigcup_{k=1}^{\infty}A_k,$$ then there exists an $n$ s.t. $\stackrel{\circ}{\overline{A_n}}\neq\emptyset$. However, is it possible ...
4
votes
1answer
78 views

Question about the Ascoli-Arzelá Theorem proof

Ascoli-Arzelá Thoerem: Let $K$ be a compact space and $M$ be a metric space and $C(K,M)$ be the set of continuous functions from $K$ to $M$. $H \subset C(K,M) $ is relatively compact if and only if ...
0
votes
1answer
50 views

Proof that uniform topology is finer than compact convergence topology.

I've tried a few approaches for the last few hours but nothing really works. I already proved that the compact convergence topology is finer than the pointwise convergence topology, if this helps. To ...
2
votes
1answer
25 views

Show that the image of a complete metric space under a continuous map is also complete given an additional condition.

This is a problem from revision material for a functional analysis class. Let $(X,d)$ and $(C,p)$ be two metric spaces and let $f:X\rightarrow C$ be a continuous function with $f(X)=C$. Assuming ...
3
votes
3answers
186 views

Prove that discrete metric space is complete

I understand the proof but I want to confirm one. So in discrete metric space, every Cauchy sequence is constant sequence and that way every Cauchy sequence is convergent sequence. Thus we conclude ...
0
votes
2answers
54 views

Doubt on Arzela-Ascoli theorem

Consider a sequence of equicontinuous and uniformly bounded functions on a compact set. Under which condition I can say that it has a unique uniformly convergent subsequence ? Or, atleast uniform ...
2
votes
1answer
27 views

Approximation of $f\in C[a,b]$ by functions constant on intervals of length $(b-a)/2^n$

I read (p. 405 here) that a continuous function on $[a,b]$ can be arbitrarily approximated according to the distance of space $L^2[a,b]$ defined by $d(f,g):=\|f-g\|_2=\int_{[a,b]}|f-g|^2d\mu$ with ...
0
votes
2answers
23 views

A metric capped at a maximum value is a metric

Suppose $d$ is a metric on the set $X$ and $R$ is a real number with $R>0$. For $x$, $y \in X$ define the function $d_R$ by: $$ d_R(x,y) = \begin{cases} d(x,y) & \text{if } d(x,y) \leq R \\ R ...
0
votes
1answer
15 views

Maximum of two metrics is a metric

Let $X$ be a set endowed with two metrics $d_1$ and $d_2$ and for all $x$, $y \in X$ define the function $d(x,y) = \max\{d_1(x,y),d_2(x,y)\}$. Show that $d$ is a metric on $X$. (Note I put up this ...
1
vote
1answer
17 views

What is a good simple definition or characterization of 'polyhedral space'?

I would like to give a definition of polyhedral space in $\mathbb{R}^n$ that is easy to understand by people that has some Maths knowledge but are neither expert in Calculus nor any other Mathematics ...
0
votes
1answer
49 views

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) ...
2
votes
3answers
67 views

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 ...
1
vote
2answers
62 views

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 $X$ ...
3
votes
1answer
99 views

$A$ 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 $Y$ ...
1
vote
1answer
40 views

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 ...
0
votes
1answer
60 views

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) = ...
0
votes
1answer
42 views

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 ...
1
vote
1answer
36 views

Use of Banach-like Covering theorem

where I cannot see how the highlighted equation has been obtained. I cannot understand how the setminus operation has been justified.
3
votes
3answers
45 views

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?
0
votes
0answers
20 views

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 ...
1
vote
2answers
24 views

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 ...
0
votes
2answers
21 views

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 ...
1
vote
1answer
36 views

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 ...
0
votes
2answers
30 views

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 ...
1
vote
2answers
27 views

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.
2
votes
1answer
51 views

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 ...
0
votes
0answers
24 views

About the cardinality of a perfect set in a separable metirc space

Let $(X,d)$ be a separable metric space, and let $E \subset X$, suppose $E$ is a perfect set (i.e. $E$ is closed and every point of $E$ is a limit point of $E$); in other words, suppose that $E = ...
0
votes
3answers
66 views

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 ...
0
votes
1answer
34 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 ...
4
votes
1answer
48 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 ...
1
vote
2answers
49 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 $|x_0 - y|<\epsilon$.
0
votes
1answer
64 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 $x$, $y \in X$. ...
2
votes
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 ...
1
vote
2answers
63 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 ...
2
votes
1answer
85 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 ...
1
vote
3answers
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 ...
0
votes
1answer
40 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)| + ...
2
votes
1answer
31 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 ...
1
vote
0answers
25 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 ...
1
vote
1answer
55 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 ...
0
votes
0answers
41 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 ...
1
vote
2answers
39 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 ...
0
votes
1answer
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 ...
2
votes
1answer
73 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 ...