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.

learn more… | top users | synonyms (1)

0
votes
0answers
5 views

Proving $(C_b(X,Y),d_{\infty})$ is complete if $(Y,d_y)$ is complete

Let $(X,d_X)$ and $(Y,d_Y)$ be metric spaces. We say a function $f: X \rightarrow Y$ is bounded if $f(X)$ is a bounded set of $Y$. Consider $$C_b(X,Y) = \left\{f: X \rightarrow Y \mid f \ \text{is ...
0
votes
0answers
20 views

Showing $(l^1(\mathbb{N}), d_1)$ is complete

Let $$ l^1(\mathbb{N}) = \left\{(x_n)_n \mid \sum_{n=0}^{\infty} |x_n| \ \text{converges} \right\}. $$ Prove that $(l^1(\mathbb{N}), d_1)$ with $d_1(x,y) = \sum_{n=0}^{\infty} |x_n - y_n|$ is a ...
1
vote
0answers
14 views

Every metric $\delta$ on sets is $\delta (A, B) = \mu(A \Delta B)$, resp. $\delta(A, B) = \mu(A \Delta B)/\mu (A \cup B)$, for some measure $\mu$?

Suppose $\mu$ is a measure and that $\delta (A, B) = \mu(A \Delta B)$ (where $\Delta$ represents symmetric difference, and $A=B$ whenever $\mu(A \Delta B) = 0$). Then $\delta$ is a distance function. ...
2
votes
0answers
37 views

How do I prove that the compact open topology is metrizable?

Reference: Conway - Functions of one complex variable Let $G$ be open in $\mathbb{C}$ and $\{K_n\}$ be a sequence of compact subsets of $G$ such that $\bigcup_n K_n = G$ and $K_{n}\subset Int(K_{n+...
1
vote
1answer
65 views

Jaccard dissimilarity and the triangle inequality

Suppose that $\delta(A, B) = \dfrac{A \Delta B}{A \cup B}$, where $\Delta$ represents symmetric difference. Then how does one prove the triangle inequality, viz that $\delta(A, B) + \delta(B, C) \ge \...
1
vote
0answers
22 views

Continuous functions $f: X \rightarrow Y$ with trivial metric on $Y$

Let $(X,d)$ be a connected metric space. Let $(Y, d_{triv})$ be a metric space, with the trivial metric defined on it. What do the continuous functions $f: X \rightarrow Y$ look like? If $f: X \...
1
vote
1answer
24 views

Uniformly Cauchy sequence of functions

I am trying to show the following: For each $n \in \mathbb N$, let $f_n:X \to Y$, where $(Y,d)$ is a complete metric. Suppose that for every $\epsilon>0$, there exists $n_0 \in \mathbb N$ such ...
1
vote
2answers
227 views

Boundary of $M_rp$ not equal to the sphere of radius $r$ at $p$?

My problem is to find a metric space in which the boundary of $M_rp$, where $M_rp = \{q \in M: d(p, q) < r \}$, is not equal to the sphere of radius $r$ at $p, \{x \in M: d(x, p) = r\}$. ...
1
vote
2answers
32 views

Showing $f(a) \in V$ but $f(x_n) \notin V$ for every $n$

Let $(X, d_X)$ and $(Y, d_Y)$ be metric spaces and let $f: X \rightarrow Y$ be a function. Let $a \in X$ and suppose $f$ is not continuous in $a$. Prove that there exists an open subset $V$ in $Y$ ...
2
votes
0answers
36 views

$A,B$ be countable dense subsets of $\mathbb R$ , let $A,B$ be given usual subspace topologies , then there exists a homeomorphism $f:A \to B$?

Let $A,B$ be countable dense subsets of $\mathbb R$ (with usual euclidean topology ) let $A,B$ be given usual subspace topologies , then is it true that there exists a homeomorphism $f:A \to B$ ?
1
vote
0answers
40 views

Is this metric space normable?

Given the function $\rho:\mathbb{R}^n\to\mathbb{R}_+$ defined by $$\rho(x)=\log\left(\frac{\sum\left|x_i\right| e^{\left|x_i\right|}}{W\left(\sum\left|x_i\right| e^{\left|x_i\right|}\right)}\right)$$ ...
-1
votes
2answers
2k views

Space of continuous functions with compact support dense in space of continuous functions vanishing at infinity

How can we prove that the space of continuous functions with compact support is dense in the space of continuous functions that vanish at infinity?
1
vote
3answers
34 views

Show that anti-metric space can only have one point

Let's define new object. Given $X$ a set: Let anti-metric be defined as: $b: X\times X \to \mathbb{R}$ such that: $b(x,y)\ge 0, \thinspace \forall x,y \in X$ $b(x,y)=0\Rightarrow x=y$ $b(x,y) = b(y,...
1
vote
1answer
28 views

Urysohn Metrization Theorem contradiction (uniform topology homeomorphic to product topology)?

The theorem states that if $F$ is regular and has a countable basis, then it is metrizable. In Munkres' proof of this theorem, he gives a function (homeomorphism) $F:X \rightarrow [0,1]^\omega$ that ...
2
votes
1answer
91 views

Can a metric space over integers induce a topology?

Questions to get a better grasp of basic topology: A metric space is an ordered pair $(M,d)$ where $M$ is a set and $d$ is a metric on $M$, i.e., a function $$ d \colon M \times M \to \...
11
votes
3answers
2k views

What is the motivation of Levy-Prokhorov metric?

From Wikipedia Let $(M, d)$ be a metric space with its Borel sigma algebra $\mathcal{B} (M)$. Let $\mathcal{P} (M)$ denote the collection of all probability measures on the measurable space $(...
0
votes
2answers
32 views

Showing this set $A$ is closed, bounded and not compact?

Let $$ l^1(\mathbb{N}) = \left\{ (x_n)_n \mid \sum_{n = 0}^{\infty} | x_n | \ \text{converges} \right\}, $$ the space of all sequences whose associated series converge absolutely. On this space we ...
0
votes
0answers
22 views

Generalizing norms: leaving out absolute homogeneity

Given a function $\rho:X\to\mathbb{R}$ on a vector space $X$ which satisfies the following properties: $\rho(x)=0$ if and only if $x=0$ $\rho(x+y)\leq\rho(x)+\rho(y)$ $\rho(-x)=\rho(x)$ for any $...
1
vote
1answer
44 views

Alternative proof: show that any metrizable space $X$ is normal - Part 1

There is a proof online that shows that all metric spaces are normal. The proof is as follows However, it has the additional baggage of needing to show that $d(x,A)$ is continuous and $U,V$ are ...
0
votes
1answer
625 views

Uniform convergence of a sequence of functions and equicontinuity

Let $(X,d)$ be a compact metric space. I would like to prove that if $(f_n)_{n \in \mathbb{N}}$ is a sequence of continuos functions $f_n:X \to Y$ that converge uniformly in $X$, then $(f_n)_{n \in \...
2
votes
1answer
84 views

Give example of $f$ that is open but neither closed not continuous (in 2D).

I'm trying to teach my self topology. The book I'm using has the following problem: Give an example of two subsets $X,Y \subseteq \mathbb R ^2$, both considered as topological spaces with their ...
0
votes
1answer
42 views

What subbase generates metric topology?

Let metric topology be the topology generated by metric balls of a metrizable space $X$ Is there a subbase $S$ that generates the metric topology? I am asking because in most textbooks, it seems ...
3
votes
1answer
783 views

Pearson correlation and metric properties

Assuming that the data set was $z$-standardized to zero mean and unit variance (also assuming that it does not contain constant vectors). Then Pearson's r reduces to Covariance: $$\rho(X,Y) := \frac{...
0
votes
0answers
28 views

Explicit construction of an $\epsilon$ net covering

Suppose $X$ is a compact space. In particular $X$ is totally bounded and there exists $x_1,..,x_n$ such that $$ X = \bigcup_{i=1}^n U(x_i, \epsilon) $$ where $U$ is the Open Ball centered at $x_i$ ...
0
votes
1answer
18 views

Proving $\mathbb{R}^2$ is not separable for this metric?

Let $d_S$ be a metric on $\mathbb{R}^2$ defined as follows $$ d_S(x,y) = \begin{cases} || x- y|| & \text{when} \ x \ \text{and} \ y \ \text{are linearly dependent} \\ ||x|| + || y || & \text{...
0
votes
2answers
23 views

Is the following metric topological equivalent to Euclidean metric?

Let $d_S$ be a metric on $\mathbb{R}^p$ defined as $$ d_S(x,y) = \begin{cases} || x- y|| & \text{when} \ x \ \text{and} \ y \ \text{are linearly dependent} \\ ||x|| + || y || & \text{when}\ ...
5
votes
2answers
63 views

Finding a special subsequence of any Cauchy sequence

Let $(X,d)$ be a metric space and let $(x_n)$ be a Cauchy sequence in $X$. Let $(\epsilon_n)$ be a sequence of real numbers and decrease to $0$. Show that there is a subsequence $(x_{n_k})$ of $(x_n)$ ...
1
vote
3answers
41 views

Show $d_2(f,g) = \sqrt{\int\limits_0^1 |f(x) - g(x)|^2 dx}$ is a metric on $C[0,1]$

I am surprised that this question hasn't been asked on here I need to show that $$d_2(f,g) = \sqrt{\int\limits_0^1 |f(x) - g(x)|^2 dx}$$ is a metric on $C[0,1]$ Proof: As usual, positive ...
1
vote
1answer
31 views

Convergence of finite metric spaces to an infinite one

Let $\{(M_i, d_i)\}$ be an infinite sequence of finite metric spaces, where $|M_i|$ is strictly increasing with $i$. Is there a standard definition of what it means for the sequence $\{(M_i, d_i)\}$ ...
3
votes
1answer
76 views

A set $A \subset l_1$ is compact if and only if closed, bounded, and one other condition

A set $A \subset \ell_1$ is compact if and only if $A$ is closed and bounded and given any $\epsilon >0$, there exists $n_0$ such that $\sum_{k=n}^{\infty} |x_k| < \epsilon$ for all $n> n_0$ ...
1
vote
1answer
21 views

In a metric space $X$, a subset $S \subset X$ is relatively sequentially compact if and only if its closure $\overline{S}$ is sequentially compact.

Prove that in a metric space $X$, a subset $S \subset X$ is relatively sequentially compact if and only if its closure $\overline{S}$ is sequentially compact. The terms relatively sequentially ...
1
vote
2answers
34 views

Alternative characterization of complete metric space

Let $(X,d)$ be a metric space. It is complete if every Cauchy sequence for $d$ on $X$ is convergent. I've heard an alternative definition of completeness for $(X,d)$: it is complete iff the ...
2
votes
1answer
35 views

Show that any metrizable space $X$ is Hausdorff

I wish to show that any metrizable space $(X,\mathcal{T})$ is Hausdorff Proof attempt: Let $d$ be the metric that generates the topology on $X$. Pick two points $x,y \in X$, we wish to produce two ...
1
vote
2answers
19 views

Show that any metrizable space $X$ is regular

This is a quick follow up to another question Show that any metrizable space $X$ is Hausdorff Recall, a topological space $(X,\mathcal{T})$ is regular if we can separate any point $x$ from ...
0
votes
1answer
41 views

Can there be a metric space where no contraction has a fixed point?

We know that: If $X$ is a metric space, then every contraction has at most one fixed point. (Note: if metric space is complete, then we have existence and uniqueness) I wonder if there can be a ...
6
votes
3answers
70 views

Show that linear functional $L(f) = \int_0^1 f(x) dx$ is continuous

Let $(C[0,1], d_1)$ be a metric space of all continuous functions $f:[0,1] \to \mathbb{R}$, $d_1$ is the $L_1$ metric $$d_1(f,g) = \int\limits_0^1 |f(x) - g(x)| dx$$ Show that linear functional $L(...
-1
votes
2answers
15 views

Locus in $(\mathrm R ,d_{\infty})$ with $d_\infty(x,y)=\max\limits_i |x_i-y_i|$ [closed]

Find the locus of points $(x_1,x_2)$ in the plane such that their distance from $(1,2)$ is equal to $3$ at $(\mathrm R ,d_\infty(x,y)=\max\limits_i |x_i-y_i|)$ I have no idea how this is looks ...
1
vote
2answers
38 views

Proof that the nowhere differentiable functions are dense in $C_b(\mathbb R)$.

I tried to make a proof, where I use a Weierstrass function. I was surprised at how easy it was, and thus a little doubtful as to the correctness of the proof. I've looked it over, and didn't find any ...
2
votes
0answers
34 views

Show that if $(X, \mathcal{T}), (Y, \mathcal{J})$ are both metrizable, then $X \times Y$ with product topology is metrizable

Let $(X, \mathcal{T}), (Y, \mathcal{J})$ be both metrizable, then Claim: $X \times Y$ with product topology is metrizable I have made an attempt at this question but the notation is ...
1
vote
1answer
30 views

Determining if this mapping is continuous?

Let $X$ be a closed and bounded subset of $\mathbb{R}^p$ and let $C(X)$ denote the vector space of continuous functions from $X$ to $\mathbb{R}$. For $f,g \in C(X)$, let $$ d_{\infty} (f,g) = \sup \...
0
votes
2answers
37 views

The set of all polynoms are closed at $d(x,y)=\max\limits_{[a,b]} \mid x(t)-y(t)\mid$?

Prove or disprove with counter-example: the set of all polynoms are closed at $d(x,y)=\max\limits_{[a,b]} \mid x(t)-y(t)\mid$ The polynoms in the interval $[a,b]$ Attempt: counter-example: $y(...
1
vote
1answer
19 views

Are symmetric and $\Delta$-metric common terminologies?

In these notes on metric spaces, the author also defined something known as "symmetric", and $\Delta$-metric. I have never seen these terminologies before. Are these terms standard usage? Can ...
0
votes
2answers
29 views

$X$ be real i.p.s. dim.>1 , if two closed balls,none of which is a subset of the other,intersect then do the boundaries of the balls intersect too?

Let $X$ be a real inner product space of dimension more than $1$ , let $B[x;r] , B[y;s]$ be two closed balls having non-empty intersection where none of the balls is a subset of the other , then is ...
0
votes
2answers
40 views

$f: \mathbb{R}^{n+1}-\{0\}\rightarrow S^{n}$ via $x \mapsto\frac{x}{\|x\|}$ is continuous

I'm having trouble understanding why a map $f: \mathbb{R}^{n+1}-\{0\}\rightarrow S^{n}$ (unit $n$-sphere, $n\ge 1$) via $x \mapsto\frac{x}{||x\|}$ is continuous. Since Unit n-Sphere under Euclidean ...
5
votes
3answers
50 views

Show $A\cap B \neq \varnothing \Rightarrow \operatorname{dist}(A,B) = 0$, and $\operatorname{dist}(A, B) = 0 \not\Rightarrow A\cap B \neq \varnothing$

I have a question Let $d$ be a metric on $X$, and define the set to set distance $$\operatorname{dist}(A,B) = \inf\{d(x,y): x\in A, y \in B\}$$ where $A,B \subseteq X$ are nonempty sets ...
0
votes
1answer
54 views

Diameter of set in metric space

I do agree with the statement that $$d(A) = \sup{\{d(x, y):x, y \in A\}}$$ But why can't we use maximum because according to me its max will also give diameter. I know it should not be correct, so ...
4
votes
3answers
122 views

On the separation axiom in a Lawvere or “generalized” metric space

According to the nLab, Lawvere metric spaces occur rather naturally (that is as certain enriched categories). A Lawvere metric space is a set $X$ equipped with a function $d : X\times X \to [0,\infty]$...
0
votes
1answer
25 views

If $\max\limits_{[a,b]}\mid x(t)-y(t)\mid$ bounded then $\sqrt{\int_a^b(x(t)-y(t))^2\text dt}$ bounded?

Prove or disprove with counter-example: if the set of the functions are bounded at $d(x,y)=\max\limits_{[a,b]}\mid x(t)-y(t)\mid$ then the set also bounded at $d(x,y)=\sqrt{\int_a^b(x(t)-y(t))^2\text ...
1
vote
0answers
31 views

Contractions mappings bijective maps boundarys on boundarys?

I remenber here the concept of a contraction mapping. Definition: Let (X, d) be a metric space. Then a map T : X → X is called a contraction mapping on X if there exists q ∈ [0, 1) such that $$ d(T(...
0
votes
1answer
26 views

How do you prove triangle inequality for this metric?

Let $f: \mathbb{R}^+ \rightarrow \mathbb{R}^+$ be an increasing concave function such that $f(t) = 0$ if and only if $t = 0$. Let $(X,d)$ be a metric space. Show that $d_f = f \circ d $ defines a ...