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)

2
votes
2answers
38 views

3 homeomorphisms between spaces (2 with jungle metric)

I've been studying for my final exam from topology and I found such an exercise. Let $X=([0,1]\times\{0\})\cup \bigcup_{n=1}^{\infty}(\{\frac{1}{n}\}\times[0,\frac{1}{n}])$ Let ...
3
votes
1answer
92 views

Is this (countable) product space complete?

Let $((X_n, {\rm d}_n))_{n \geq 0}$ a sequence of complete metric spaces. Suppose that all the metrics are bounded by $1$. Consider $X = \prod_{n \geq 0}X_n$ with the metric given by: $${\rm ...
0
votes
1answer
99 views

Alternate proof for Arzela-Ascoli

Im trying to finish a beautiful excercise, which consist of giving an alternate proof for the following corollary of Arzela-Ascoli´s Theorem. Given $X,Y$ metric spaces, $X$ compact, $Y$ complete, and ...
3
votes
4answers
93 views

Does there exist a path connected metric space , in which at least one open ball is countable ?

Does there exist a path connected metric space with more than one point , in which at least one open ball is countable ?
2
votes
1answer
41 views

Completion of this metric space

Let $d'$ be a metric on $C^1([0,1])$ as follows: $$ d':\ C^{1}\left([0,1]\right)^{2}:\ |f(0)-g(0)| + \sup\left\{ |f'(x)-g'(x)| \mid x\in[0,1] \right\} $$ I've already managed to prove that this is ...
6
votes
2answers
47 views

Metric in $\mathbb{P}_2$

I have to prove that $\mathbb{P}_2$ with the function $\delta(P,Q)$ defined by "Sine of the angle between two vector in $\mathbb{R}^3$ such that they correspond respectively to P and Q" is effectively ...
2
votes
2answers
66 views

Is it mathematically correct to say that if the metric is flat/curved the *shortest* path is/not a Euclidean straight line?

Is it mathematically correct to say that if the metric is flat/curved the shortest path is/not a Euclidean straight line? I am still hesitant to make this claim, due to at least one counter example. ...
3
votes
1answer
33 views

Proving that $d(f,g)=\|f-g\| = \sup \limits_{0\leq x \leq 1} |f(x)-g(x)|$ is a metric on $X=C[0,1]$

Let $X=C[0,1]$ be the set of all continuous functions on $[0,1]$. For any two functions $f,g\in X$, set $$d(f,g)=\|f-g\| = \sup \limits_{0\leq x \leq 1} |f(x)-g(x)|.$$ Prove $(X,d)$ is a metric ...
2
votes
0answers
34 views

Prove or disprove that the Bhattacharyya distance is a true distance function

Let $\mathcal{X}\equiv\Bbb{R}^n\times\Bbb{S}_{++}^n$, where $\Bbb{S}_{++}^n$ is the space of all symmetric positive-definite $n\times n$ real matrices. Let $x,y\in\mathcal{X}$, where $$ ...
3
votes
2answers
51 views

The set of irrational numbers is not a $F_{\sigma}$ set.

I want to proove that the set of irrational numbers is not a $F_{\sigma }$ set and also the set of rational numbers is not a $G_{\delta}$ set using Baire theorem. I started with saying that ...
3
votes
2answers
25 views

Complete product of metric spaces

Prove that: If a product $X\times Y$ of metric spaces $(X,\rho_X)$ and $(Y, \rho_Y)$ with metric $\rho((x_1,y_1),(x_2,y_2))=\sqrt{(x_1-y_1)^2+(x_2-y_2)^2}$ is complete, then metrics $\rho_X$ and ...
0
votes
1answer
18 views

Continuous function on interval, how do balls look.

Consider the metric space $C([a,b]),d_1$. $$ d_1:\ C([a,b])^2 \rightarrow \mathbb{R}:\ (f,g) \mapsto \int_a^b|f(x)-g(x)|\ dx$$ Is this metric space a normed vector space? How do open balls look? The ...
0
votes
1answer
11 views

Clarification of Sequential characterization of closedness of the set

I've been trying to understand $ \Leftarrow $ part of proof from link http://math.stackexchange.com/a/153372/240184 I dont understand why Hence the closure of F is a subset of F, whence they are ...
1
vote
1answer
43 views

How do you call a metric space with “continuous” points?

I have the impression that "continuous space" is not a mathematically precise concept (as opposed to continuous functions that can be defined under various contexts). However, I find I need to refer ...
0
votes
1answer
20 views

Nonincreasing and nondecreasing sequences in Hausdorff metric

For every metric space $(X,d)$ we have the Hausdorff metric space $(\mathcal{H}(X),H)$ that assosiates with it, where $\mathcal{H}(X)$ is the space of nonempty compact subsets of $X$ and $H$ is the ...
0
votes
2answers
20 views

Subsequence implicate bounded and closed set

I've been thinking about that problem for a long time, now it is right time to ask! Problem: Proof that if $ K \subset \mathbb{R}^{d} $ is such a set that every sequence with elements in $ K $ ...
4
votes
3answers
251 views

Are there infinitely many non equivalent metric spaces on certain sets (?)

Two metric spaces X and Y are called equivalent if: $d_X (x,x_n) \to 0 \Leftrightarrow d_Y (x,x_n) \to 0 $ with $ n \to \infty $ I wonder whether, if you took a certain set (for example a finite set, ...
0
votes
1answer
23 views

Question about Rudin's Functional Analysis Closed Graph Theorem

In page 51 of Rudin's Functional Analysis, the closed graph theorem is proven, which says that if you have a linear map between two F-spaces whose graph is closed in the product space, then the map is ...
4
votes
1answer
28 views

Calculate the length of $\gamma(t)=(t,t), t \in [-1,-\frac{1}{2}]$ with the metric $g=\frac{dx^2+dy^2}{y^2}$ and compare with euclidean metric

Consider the metric $g=\frac{dx^2+dy^2}{y^2}$ on $\mathbb{R}_+^2=\{(x,y) \in \mathbb{R}^2 : y>0\}$. Calculate the length of the curve $\gamma(t)=(t,t), t \in [-1,-\frac{1}{2}]$ and compare ...
1
vote
1answer
46 views

All lines of $\mathbb R^3$ are isometric to $\mathbb R$

I have just started reading Metric Spaces by Michael Searcoid. The first Chapter states a result : Suppose $n \in \mathbb N~\forall~ i \in \mathbb N_n,(X_i,\tau_i) $ is a non empty metric space. ...
0
votes
2answers
48 views

The shortest path in a metric space with a given metric

My questions seem to be very basic and intuitively correct but I can't formally prove them. Before learning metric spaces, for $R^2$, we always define the distance between 2 points as $d_2 = ...
2
votes
1answer
70 views

Is there a name of such functions?

Let $U$ be an open subset of $ \mathbb R^n$ and consider $f :\mathbb R^n \to \mathbb R$ with the properties that $ f( \partial U)=0$ and $f$ takes negative values on $U$. My questions: Is there ...
2
votes
1answer
25 views

Boundary and limit points

Suppose that $\Omega \in \mathbb{R}^n$. Prove that if $\vec{x} \notin \Omega$ and $\vec{x}$ is a boundary point of $\Omega$, then $\vec{x} $ is a limit point of $\Omega$. My try: $\vec{x}$ is a ...
0
votes
2answers
39 views

Prove that if $(f_n)_{n \geq 1} \to f$ uniformly then $f_n \to f \in B(X)$

Let $B(X)$ be the set of bounded functions from $X$ to $\mathbb{C}$ where $X$ is any metric space. Let $(f_n)_{n \geq 1} $ be a sequence in $B(X)$. Show that if $f_n \to f$ uniformly where ...
3
votes
5answers
174 views

A finite set is closed

Question: Prove that a finite subset in a metric space is closed. My proof-sketch: Let $A$ be finite set. Then $A=\{x_1, x_2,\dots, x_n\}.$ We know that $A$ has no limits points. What's next? ...
2
votes
4answers
49 views

Transferring the usual metric from $\mathbb{R}$ to $(0, 1)$ gives us a complete metric space on $(0, 1)$?

I'm watching this video here - https://www.youtube.com/watch?v=zcAvVTFUxS8 The lecturer says that $\mathbb{R}$ and $(0, 1)$ under the usual metric are homeomorphic yet $\mathbb{R}$ is complete and ...
1
vote
1answer
18 views

Examples of uniformly discrete proper metric spaces which are not countable.

By uniformly discrete I mean there exists a $C > 0$ such that for all $x \neq y$ we have $d(x, y) \geq C$. By proper I mean the preimage of every closed ball is compact. Are there any examples of ...
0
votes
1answer
27 views

Proof of triangular inequality for $|z_1-z_2|$

I want to prove that the following is a metric on $\Bbb C$. $|z_1-z_2|$ where $z_1,z_2\in \Bbb C$. I have done all easily but triangular inequality: For triangular inequality, I applied it, as if it ...
1
vote
1answer
48 views

Prove that the function in $[0,\pi]$ defined by $f(x)=\sin(x)/x$ and $f(0)=1$ is a contraction

Let $f$ be a function $f:[0,\pi]\to\mathbb{R}$ such that: $$f(x)=\left\{ \begin{array}{ll} \frac{\sin(x)}{x} & \mbox{if } x \neq 0 \\ 1 & \mbox{if } x = 0 \end{array} \right.$$ I want ...
1
vote
1answer
23 views

For each $n \in \mathbb{N}$ consider $A_n = \{1/n\}\times [0,1]$ and let $X=\bigcup_{n \in \mathbb{N}}A_n \cup \{(0,0)(0,1)\}$

For each $n \in \mathbb{N}$ consider $A_n = \{1/n\}\times [0,1]$ and let $X=\bigcup_{n \in \mathbb{N}}A_n \cup \{(0,0)(0,1)\}$. Show that $\{(0,0)\}$ and $\{(0,1)\}$ are connected components. ...
1
vote
1answer
22 views

Is this a metric on the shift space?

Consider $$ \Omega_N:=\left\{\omega=(\ldots,\omega_{-1},\omega_0,\omega_1,\ldots): \omega_i\in\left\{0,1,\ldots,N-1\right\}\text{ for }i\in\mathbb{Z}\right\} $$ and $$ ...
12
votes
1answer
240 views

Let $D$ be a bounded domain (open connected) in $ \mathbb C$ and assume that complement of $D$ is connected.Then show that $\partial D$ is connected

I am trying to prove the following famous result in Point Set Topology. Let $D$ be a bounded domain (open connected) in $ \mathbb C$ and assume that complement of $D$ is connected. Then show that ...
1
vote
1answer
28 views

What does $\overline B(0,1)\subseteq (\mathscr{C}([0,1],\mathbb{C}),d_\infty)$ mean?

What does $\overline B(0,1)\subseteq (\mathscr{C}([0,1],\mathbb{C}),d_\infty)$ mean? Could someone explain the meaning of this? How can a closed ball be a subset of a metric space of continous ...
4
votes
1answer
64 views

Generalization of metric spaces?

Usually we define a metric on a space $X$ to be a map $X\times X\to\mathbb{R}$ that satisfies a few axioms. $\mathbb{R}$ has of course a total order. What if we instead have a metric $X\times X\to A$ ...
3
votes
1answer
68 views

Is the completion of a metrizable topological group metrizable?

Let $G$ be a topological group and its two-side uniformity $\mathcal{U}$ (that is the uniformity generated by right uniformity and left uniformity of $G$) coincides with the uniformity of a metric ...
0
votes
1answer
37 views

Show that a metric space has a countable base if and only if each open cover has a countable subcover

Show that a metric space has a countable base if and only if each open cover has a countable subcover. Proof $\Rightarrow$ using Lindöf's lemma But how does one show $\Leftarrow$? Consider $\{ ...
3
votes
2answers
23 views

Locally connected metric spaces

I have the following definition of a locally connected metric space. Given $(X,d)$ a metric space, $x \in X$ and given $U \ni x$ a neighbourhood. Then there exist a connected neighbourhood $V$ susch ...
1
vote
1answer
50 views

Error in the reasoning?

Give an example of a metric space $(M,d)$, an $a\in M$ and a $R\in \mathbb{R}, R> 0$ such that $$\overline{B(a,R)} \not = \overline{B} (a,R)$$ Prove in general ...
2
votes
1answer
36 views

Every metric space contains a discrete, coarsely dense subset

I'm wondering on how to prove the following: Let $(X,d) $ be a metric space. We say that a subset $ A \subset X $ is coarsely dense iff $ \exists_{C > 0} \forall_{x \in X} \exists_{a \in A} d(x,a) ...
4
votes
3answers
65 views

Closest packing of equal balls in $\Bbb{R}^4$

I know how to find the closest packing of equal spheres in $\Bbb{R}^3$. I'd like to know how to find the closest packing of equal balls in $\Bbb{R}^4$ with the standard Euclidian metric. I suspect ...
0
votes
0answers
22 views

Prove that $\frac{d(a,b)}{1 + d(a,b)}$ is a metric? [duplicate]

Given any metric $d(a,b)$ where $a,b\in\mathbb{R}$, prove that $$d_1(a,b):=\frac{d(a,b)}{1+d(a,b)}$$ is also a metric. We have $0 \leq s\leq t \Rightarrow \frac{s}{1+s} \leq \frac{t}{1+t}$ to support ...
0
votes
2answers
22 views

Open set in a metric space is union of closed balls?

We know that every open set $A$ is in a metric space $(X,d)$ is the countable union of closed sets, and every open set $A$ is in a Euclid space $R^n$ is the countable union of closed balls. My ...
1
vote
1answer
26 views

Inequality involving distance between two points

Let $\Omega$ be a set in $\mathbb{R}^n$ Fix $x\in\Omega$, and $p\notin \Omega$ Then, is it always true that $$\|x-p'\| \leq \|x-p\|$$ , where $\|\cdot\|$ is the Euclidean norm, and $p'$ is the ...
0
votes
2answers
45 views

Show that $\mathbb{R}$ and $\mathbb{R}^n$ are not homeomorphic if $n\geq 2$

Show that $\mathbb{R}$ and $\mathbb{R}^n$ are not homeomorphic if $n\geq 2$. I want to use a connection-type argument. I thought of giving the following proof; Suppose that there exist such a ...
1
vote
1answer
30 views

The space of absolutely convergent series is complete

For clarity: $$ l^{1}(\mathbb{N}) = \left\{ (x_{n})_{n} \ \middle|\ \sum_{n}|x_{n}| \in\mathbb{R} \right\} $$ $$ d_{1}:\ l^{1}(\mathbb{N}) \times l^{1}(\mathbb{N}) \rightarrow \mathbb{R}^{+}:\ ...
0
votes
1answer
20 views

How to convert table into a distance function?

Been stumped on this past paper question for a while, it's in the context of clustering and the next part is using single linkage bottom-up hierarchical clustering to form a dendrogram using your ...
0
votes
1answer
38 views

Formal name for the coordinate values of the pushforward of the inverse metric on an embedded manifold?

What is the formal name of the following object: \begin{align}\tag{4} \Delta^{\alpha \beta} = \dfrac{\partial y^\alpha}{\partial x^m} g^{mn} \dfrac{\partial y^\beta}{\partial x^n} \end{align} where ...
0
votes
1answer
13 views

A convergent sequence of non-expansions converges uniformly on a totally bounded domain

Here's a theorem that I tried to prove: Let $V,d_V$ and $W,d_W$ be metric spaces and $(f_n)_n$ a sequence of non-expansions that converges to a function $f:A \subseteq V \rightarrow W$: $$ f_n:\ A ...
1
vote
1answer
35 views

Can I pull a limit through a metric?

Let $X,d_X$ and $Y,d_Y$ be metric spaces and $(f_n)_n$ a sequence of (continuous) functions. Does this hold and, more importantly, why? $$ d_Y\left(\lim_{n\rightarrow +\infty}f_n(x), ...
0
votes
1answer
25 views

Book recomendation for function sequences.

I wanted to study about sequences of functions defined in metric spaces. What book/books do you recommend? Thanks!