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)

0
votes
1answer
25 views

How do you prove that a metric space $X$ is separable if and only if $X$ has a countable subset $Y$ with property below?

A metric space $X$ is separable if and only if $X$ has a countable subset $Y$ with property: for $\epsilon > 0$ and every $x \in X$, there is a $y \in Y$ such that $d(x, y) < \epsilon$.
2
votes
3answers
106 views

Open Sets in $\mathbb{R}$

I was wondering what the general form of an open set is in the real numbers. Is it just an interval of the form $(a,b)$; $a,b \in \mathbb{R}$.
1
vote
1answer
33 views

If $E$ is closed and bounded in a metric space $X$ and $f: E → R$ is continous on $E$ , prove that $f$ is bounded on $E$.

If $E$ is closed and bounded in a metric space $X$ and $f: E → R$ is continous on $E$ , prove that $f$ is bounded on $E$. and suppose that $X$ satisfy the Bolzano Weierstrass Property attempt: ...
2
votes
0answers
41 views

The projection map $\pi_j : \Bbb R^n \to \Bbb R$ given by $\pi_j(x) = x_j$ is open but not closed.

The projection map $\pi_j : \Bbb R^n \to \Bbb R$ given by $\pi_j(x) = x_j$ is open but not closed. I have found an example for the map not to be closed. But unable to prove that it is open. Please ...
8
votes
1answer
100 views

Characterization of the circle within metric spaces

There are various characterizations of the circle. To be precise, there is not the circle. There are several categories which contain an object we refer to as "the circle". In $\mathsf{Top}$ the ...
1
vote
1answer
30 views

Doubt related to the extended real line and distance/metric

I am studying real analysis and I am being introduced to functions that take values on the extended real line, I have a fundamental doubt about this so I'll give an example to illustrate my confusion: ...
2
votes
2answers
21 views

If $g: Y \to Z$ is a continuous injection, then a map $f : X \to Y$ is open if $g\circ f$ is open.

If $g: Y \to Z$ is a continuous injection, then a map $f : X \to Y$ is open if $g\circ f$ is open. To show the map $f : X \to Y$ is open, we first take any open subset $U$ from $X$ and then show that ...
2
votes
0answers
34 views

The square $S := [- R, R] \times [-R, R]$ is a compact subset of $\Bbb R^2$.

The square $S := [- R, R] \times [-R, R]$ is a compact subset of $\Bbb R^2$. An intuitive approach: Let $S$ be not compact then there is an open cover of which there is no finite sub cover of $S$.Now ...
0
votes
1answer
28 views

Sequence characterization of bounded sets

If $M$ is an arbitrary metric space, the following holds: $A\subseteq M$ is totally bounded $\Leftrightarrow$ Each sequence in $A$ contains a Cauchy subsequence. Additionally, for ...
1
vote
1answer
25 views

Is $\overline{\mathbb{R}}^+$ a compact Polish space

if $X$ is defined by $$X= [0,+\infty)\cup\{+\infty\}$$ is endowed with the metric $$d_X(x,y) = |\arctan(x) - \arctan(y)|$$ Is it true that the metric space $(X,d_X)$ meets the following properties? ...
0
votes
1answer
26 views

Alternate definition limit

The definition of the limit of a sequence is: $L=\lim\limits_{n\to\infty}f(n)\Leftrightarrow\forall\epsilon>0:\exists N\in\mathbb{N}:\forall n\in\mathbb{N}:\left(n>N\Rightarrow ...
1
vote
0answers
33 views

Name of the metric: $d(f,g)=\max \limits_{a\leq x \leq b} |f(x)-g(x)|$

What is the name of the metric: $$d(f,g)=\max \limits_{a\leq x \leq b} |f(x)-g(x)|$$ Where $f,g\in X$ where $X$ is the space of all continuous functions. I can't find any documentation on this ...
1
vote
1answer
28 views

Checking my understanding of the Interior of these intervals

Let $[a,b]$ be any finite closed interval. (i) $\text{Int}_{[a,b]}(a,b]$ Am I correct to say that the interior of this set is $[a,b]$? Since the interior of a set are all the points in the set in ...
0
votes
1answer
33 views

An example of a dense and co-dense set in a metric space with countable derived set

Let $(X,d)$ be a metric space and $A\subset{X}$ such that $A$ and $A^c$ are both dense in $X$. Show that it is not necessary that $A^\prime$ be uncountable. And prove $(A^\prime)^\prime=A^\prime$. ...
0
votes
2answers
26 views

If $A$ is connected , $B$ is open and closed and $A \cap B$ is non empty then $A \subset B$.

Let $(X,d)$ be a metric space and $A,B \subset X$. If $A$ is connected , $B$ is open and closed and $A \cap B$ is non empty then $A \subset B$. I tried it with proving a contradiction if we first ...
0
votes
0answers
22 views

Show that a point $ a \in X$ is not a cluster point of $X$ if $a$ is isolated.

A point $a$ in a metric space $X$ is said to be isolated if and only if $r> 0$ so small that $B_r(a)$ = {$a$} Show that a point $ a \in X$ is not a cluster point of $X$ if $a$ is isolated. proof: ...
0
votes
1answer
34 views

A complete subset of a metric space is closed?

Supposing $A$ is a subset of a metric space $S$, it is simple enough to show that if $S$ is complete and $A$ is closed, that $A$ is complete. However, without being given that $S$ is complete, what ...
3
votes
1answer
51 views

Does there exist non-compact metric space $X$ such that , any continuous function from $X$ to any Hausdorff space is a closed map ?

I know that there is a topological space $X$ which is not compact but such that , for any Hausdorff topological space $Y$ , any continuous function $f:X \to Y$ carries closed sets to closed sets . I ...
1
vote
1answer
19 views

Let $X$ be the union of axes is it homeomorphic to a line, a circle, a parabola or the rectangular hyperbola $xy = 1$?

Let $X$ be the union of axes given by $xy = 0$ in $\Bbb R^2$ . Is it homeomorphic to a line, a circle, a parabola or the rectangular hyperbola $xy = 1$? If we remove the origin from the union of axes ...
0
votes
1answer
17 views

Evenly Spaced Integer Topology is Metrizable

Fustenborg's proof uses an evenly spaced integer topology on $\mathbb Z$ which declares that a basis of open sets as those of the form $a + b \mathbb Z$ (i.e. arithmetic progressions). I'm interested ...
1
vote
2answers
53 views

Is there a metric proof for the Caristi theorem?

I'm writing a paper about hyperconvexity in metric spaces and came across Caristi's theorem: Let $(X, d)$ be a complete metric space and $\phi\colon X \to \mathbb{R}^+$ a continuous function. An ...
1
vote
1answer
34 views

How many degrees of freedom are in a flat metric and how does one count them?

I think that there are zero degrees of freedom in a flat metric, but I do not know how to count them. I know that any symmetric metric tensor has $n(n-1)/2$ degrees of freedom, where $n$ is the ...
0
votes
1answer
28 views

How to find an open ball for a metric space?

I don't understand the process to find the open ball. I understand the definition and I understand that for B(0, delta), I need to substitute x as 0. After this stage, I don't understand where to go ...
1
vote
2answers
34 views

Help needed in a proof to show path connected-ness of a special kind of set in real Euclidean space

Let $A$ be a connected subset of $\mathbb R^n$ and $\epsilon >0 $ , consider $V:=V(\epsilon , A):=\{x \in \mathbb R^n : d_A(x)< \epsilon \}$ , I need to show that $V$ is path connected , so we ...
2
votes
3answers
54 views

Is every open cover of $\{(x,y) \in \mathbb R^2 : x^2+y^2=1\}$ also an open cover of some $\delta$ annulus around the unit circle?

Let $\{U_\alpha\}$ be an open cover of $\{x \in \mathbb R^n:\|x\|=1 \}$ , $n \ge 2$ , then does there exist $\delta >0$ such that $\{U_\alpha\}$ is also an open cover of $\{ x \in \mathbb R^n : ...
1
vote
1answer
28 views

$f : U \to \Bbb R$ be a differentiable function such that $D f (p) = 0$ for all $p \in U$. Then $f$ is a constant function.

Let $U$ be an open connected subset of $\Bbb R^n$ and $f : U \to \Bbb R$ be a differentiable function such that $D f (p) = 0$ for all $p \in U$. Then $f$ is a constant function. I am facing ...
2
votes
0answers
32 views

Given any two points $x, y \in X$ there exists connected set $A$ such that $x, y \in A$. Then $X$ is connected.

Let $X$ be a (metric) space such that given any two points $x, y \in X$ there exists connected set $A$ such that $x, y \in A$. Then $X$ is connected. Let us consider a continuous function $f : X \to ...
1
vote
1answer
26 views

Completion of a sequence space

Let $F$ be a field with some absolute value $|\cdot|$. Consider the space $X$ of sequences $\mathbf{a} = (a_1, a_2, a_3, \cdots)$ for which $a_i \in F$ for all $i\in\mathbb{N}$ and at most finitely ...
0
votes
1answer
32 views

Distance on riemann sphere [duplicate]

Let we have $C$ the set of complex numbers and $z_1 , z_2 \in C $ we have $Z_1 , Z_2 \in S$ correspond on riemann sphere and we will define : $$ d(Z_1,Z_2)=\frac{2|z_1-z_2|}{\sqrt{1+|z_1|^2} ...
2
votes
3answers
30 views

Continuity over a compact subset of a metric space implies continuity everywhere

Let $f: (X, d_X) \rightarrow (Y, d_Y)$ be a function from metric spaces. If $f$ restricted to any compact subset of $X$ is continuous, then $f$ must be continuous everywhere. Should I proceed with ...
0
votes
0answers
83 views

Prove that $d((x|y),(u|v)) = max \{d_{x}(x|u),d_{y}(y|v)\}$

$d_{x}$ is a metric on the set $X$. $d_{y}$ is a metric on the set $Y$. Prove that $$d((x|y);(u|v)) = max \{d_{x}(x|u),d_{y}(y|v)\}$$ defines a metric on the set $X \times Y$. I did the following: ...
1
vote
3answers
51 views

Let $A$ be a connected subset in $ \Bbb R^n$ and $ \epsilon >0$ then the $ \epsilon$ -neighbourhood of $A$ is path-connected.

Let $A$ be a connected subset in $ \Bbb R^n$ and $ \epsilon >0$ then the $ \epsilon$ -neighbourhood of $A$ defined by $U_\epsilon(A) := \{x \in \Bbb R^n : d_A(x) < e\}$ is path-connected. If ...
1
vote
1answer
55 views

Is $\{(x,y) \in \mathbb R^2 : xy=0 \}$ homeomorphic to $\mathbb R$?

Is $\{(x,0) : x \in \mathbb R \} \cup \{(0,y) : y \in \mathbb R \}$ homeomorphic to $\mathbb R$ ? I am totally stuck and I don't even have any intuition whether they should be homeomorphic or not . ...
1
vote
2answers
30 views

Let $X$ be the union of open disk in $\Bbb R^2$ along with the tangent line $x =1$ then X is connected.

Let $X$ be the union of open disk in $\Bbb R^2$ along with the tangent line $x =1$ then X is connected. Here I use the following criterion for $X$ to be connected: A metric space $(X,d)$ is ...
3
votes
1answer
44 views

Help creating a more insightful proof looking at closures of a metric space

My lecture notes from my metric space course contained the following practice questions. I am getting very confused by this question because I found the following statement on wikipedia "A metric ...
1
vote
1answer
29 views

Question about convergence in a metric space

For part a) my strategy was showing that since E is sequentially compact, by the Borel-Lebesgue theorem it is compact. For part b) I am not sure how to solve the problem. Can I simply use the ...
0
votes
1answer
21 views

$f: \mathbb R \to \mathbb R$ as $f(x)=\sin (1/x) , \forall x >0 ; f(x)=0 , \forall x \le 0$ , is the graph of $f$ connected in $\mathbb R^2$?

Consider the function $f: \mathbb R \to \mathbb R$ as $f(x)=\sin (1/x) , \forall x >0 ; f(x)=0 , \forall x \le 0$ , then $f$ is not continuous on $\mathbb R$ . Is the graph of $f$ i.e. $G(f) :=\{ ...
0
votes
1answer
27 views

Equivalent distances

I am interested in the following property about distances: Given two distances $d_1$ and $d_2$, $$ d_1(x,y_1) < d_1(x,y_2) \Leftrightarrow d_2(x,y_1) < d_2(x,y_2). $$ Under my point of view, ...
1
vote
0answers
22 views

Embedding of $K_{2,3}$ into $\ell_1$

I am looking for hints for the following problem: Prove that every embedding of $K_{2,3}$ (with the shortest path metric and unit edge-length) into $\ell_1$ has distortion at least 4/3! Notation: ...
1
vote
0answers
34 views

A measure on the space of probability measures

I've been reading about optimal transport and it's connections to geometry. At some point one has to study a bit of the structure of the space of probability measures, $\mathcal{P}(X)$, (over a metric ...
0
votes
1answer
32 views

Show that $R$ is closed but not sequentially compact.

Show that $R$ is closed but not sequentially compact. Attempt: A subset E of a metric space X is said to be sequentially compact if and only if every sequence $x_n \in E$ has a convergent ...
1
vote
1answer
50 views

Every sequentially compact set is closed and bounded.

A subset $E$ of $X$ is said to be sequentially compact if and only if every sequences $x_n \in E$ has a convergent subsequence whose limit belongs to $E$. Prove that every sequentially set is closed ...
0
votes
0answers
32 views

Concept of Boundedness

I noticed there are two notions of boundedness, one in the context of order theory and other in the context of metric spaces. In a metric space (X,d) , we talk about subsets of X being bounded iff ...
1
vote
3answers
61 views

If $\operatorname{id}:(X,d_1) \to (X,d_2)$ is continuous for any two metrics $d_1$ and $d_2$, then what will be $X$?

Let $X$ be a set with the property that for any two metrics $d_1$, and $d_2$ on $X$, the identity map $\operatorname{id} : (X, d_1) \to (X, d_2)$ is continuous. Which of the following are true? ...
1
vote
1answer
41 views

Given two balls and a point show there radii $c,d$ such that $B_c(x) \subseteq B_r(a) \cap B_s(b) $

Show that given two balls $B_r(a)$ and $B_s(b)$, and a point $x \in B_r(a) \cap B_s(b)$, there are radii $c$ and $d$ such that $B_c(x) \subseteq B_r(a) \cap B_s(b) $ and $B_d(x) \supseteq B_r(a) ...
1
vote
1answer
33 views

Show that the metric space C[a,b] is complete. [duplicate]

Prove that the metric space $C[a,b]$ is complete. Where $C[a,b]$ is the collection of continuous $f:[a,b] → R$ and $||f|| = sup_{x \in [a,b]} |f(x)|$, such that $\rho (f,g) = ||f - g||$ is a metric ...
0
votes
2answers
33 views

Let $U$ be an open connected subset and $f : U \to \Bbb R$ be a diff function then $f$ is a constant function.

Let $U$ be an open connected subset of $\Bbb R^n$ and $f : U \to \Bbb R$ be a differentiable function such that $D f (p) = 0$ for all $p \in U$ then $f$ is a constant function. If we can prove that ...
0
votes
1answer
15 views

Surjectivity of isometry

I am looking for the proof Prove of "any isometry S is a surjective mapping". My attempt: pick any two points $A, B$, consider their images $S(A) = A'$ , $S(B) = B'$ . To prove surjectivity, I need ...
2
votes
1answer
76 views

If $id:(X,d_1)\to (X,d_2)$ is continuous then what will be $X$?

Let, $id:(X,d_1)\to (X,d_2)$ is continuous. Then which is(/are) TRUE ? (A) $X$ must be singleton. (B) $X$ can be any finite set. (C) $X$ can NOT be infinite (D) $X$ may be infinite but NOT ...
0
votes
1answer
22 views

Proving a homeomorphism when graph of function has product topology

Suppose $f : (X,d_x) \rightarrow (Y,d_y)$ is a function between metric spaces, and $X \times Y$ has the product topology. The graph $G_f$ is the subspace $G_f = \{(x,f(x)) \mid x \in X\}$. Define ...