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
1answer
41 views

A point in a closed set in Euclidean Space [duplicate]

''There exists a point in a closed set which is at minimum distance from a point not in the set.'' I have no idea why this is true. Any help will be appreciated.
0
votes
1answer
15 views

If $d_1$, $d_2$ are metrics on $X$ find a relationship between $\tau_1$ and $\tau_2$.

Suppose $d_1$, $d_2$ are metrics on $X$ and whenever $x_n \rightarrow x$ using $d_1$ we have that $x_n \rightarrow x$ using $d_2$. Let $\tau_1$ be the collection of open sets of $(X,d_1)$ and ...
0
votes
1answer
17 views

Suppose $d_1$ and $d_2$ are equivalent metrics and $d_1$ is bounded, is $d_2$ bounded?

Suppose $d_1,d_2$ are topologically equivalent metrics on a set $X$. Suppose also that $d_1$ is bounded, that is there exists $K>0$ such that $d_1(x,y) \leq K$ for all $x,y\in X$. Does this mean ...
0
votes
2answers
33 views

Prove this is a metric, what else should I consider?

Let $C_b(\mathbb{R})$ be the space of the bounded continuous functions with values in $\mathbb{C}$ defined in $\mathbb{R}$ ($f:\mathbb{R}\rightarrow\mathbb{C}$) prove that: with $x\in \mathbb{R}, ...
0
votes
2answers
48 views

Metric space and continuity

We define a map $f:(S,d)→(S',d')$ between 2 metric spaces to be continuous at x belongs to S if for every sequence ${x_n}$ in $S$ that converges to x, the sequence {f(x_n)} in $S'$ is convergent to ...
0
votes
2answers
49 views

Why does this proof work: Closed unit ball in $C_0$ is not compact

I know that this question has been asked to death, and multiple solutions are given, but I still don't understand why the "standard" proof works Following Show that the closed unit ball $B[0,1]$ in ...
1
vote
1answer
28 views

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

Following Proving that $d(f,g)=\|f-g\| = \sup \limits_{0\leq x \leq 1} |f(x)-g(x)|$ is a metric on $X=C[0,1]$ I would like to prove that the same is true for bounded functions on $[0,1]$ ...
1
vote
1answer
36 views

Every open ball of a normed vector space $E$ its homeomorph to the entire space $E$.

I have some questions about this proof that "Every open ball of a normed vector space $E$ its homeomorph to the entire space $E$.": By the example (12), we just have to consider the ball $B(0,1)$, we ...
1
vote
0answers
18 views

How is $d(af(x), af(x_o))$ and $d(f(x), f(x_o))$ related?

I wish to prove that given $f \in C_0([0,1])$ of continuous function, then $af \in C_0([0,1])$ where $a \in \mathbb{R}$ I am having trouble relating $d(af(x), af(x_o))$ with $d(f(x), f(x_o))$ So to ...
-1
votes
1answer
33 views

Is it possible that two metric spaces are metrically isomorphic but not homeomorphic.

I am trying to find an example of metric spaces $(X,d_x)$ and $(Y,d_y)$ such that they are metrically isomorphic, but not homeomorphic. I have been attempting to find one, however I have not been ...
3
votes
4answers
30 views

If $d_1,d_2$ are not equivalent metrics, is it true $(X,d_1)$ is not homeomorphic to $(X,d_2)$?

Consider the statement: If $(X,d_1)$ and $(X,d_2)$ are metric spaces and $d_1,d_2$ are not equivalent metrics, then $(X,d_1)$ is not homeomorphic to $(X,d_2)$. I think this is true, however I can't ...
4
votes
1answer
23 views

There are finitely many $n$-dimensional wallpaper groups

There are $7$ Frieze groups, $17$ wallpaper groups and $230$ space groups, which are the 1,2,3-dimensional cases of isometries on $\Bbb R^n$ (I think? Frieze groups seem to require $[0,1]\times\Bbb R$ ...
0
votes
1answer
30 views

Normalized measure over compact metric spaces

Consider the following definitions. Let $M = (V,T,d)$ be a compact metric space with finite diameter $$D = D(M) = \max d(x,y), ( x, y \in M)$$ and a finite normalized measure $\mu$$M$(.), ...
3
votes
3answers
141 views

Distance from a point to empty set.

Let $(X,d)$ be a metric space and let $A \subseteq X$. We define the distance from a point $x \in X$ to $A$ by $d(x,A)= \inf \{ d(x,a) : a \in A \} $. What will be the value of $d(x, \emptyset )$? I ...
0
votes
1answer
54 views

How can we write (2,5) in the countable family of disjoint open intervals?

I have just read a theorem which states that "Every open subset of R is the union of countable family of disjoint open intervals". Now,I want know how can we write (2,5) in the countable family of ...
1
vote
1answer
23 views

open\closed and disjoint sets under R2

I am stuck with the following question: Consider the sets in $\mathbb{R}^2$ defined by $A = \{(x,1/x)| x > 0 \}$, $B = \{(x, −1/x)| x < 0\}$. Prove that the sets are closed and disjoint, and ...
0
votes
1answer
27 views

Showing that a function is not $(d,d)-$ continuous at a point.

Let $d: \mathbb R \times \mathbb R \rightarrow \mathbb R$ be a metric: $$ d(x,y) = \begin{cases} 0 & x = y \\ |x| + |y| + 3|x-y| & x \neq y \end{cases} $$ Show that the function $f: \mathbb R ...
0
votes
1answer
23 views

Does the metric induce the topology on $X$?

Let $X=\left\{a,b,c\right\}^{\mathbb{Z}}$ and on $X$ the product topology $\tau$, where on $\left\{a,b,c\right\}$ we consider the discrete topology. On $X$, consider the metric $$ ...
0
votes
1answer
30 views

Topology in the set of matrices

Let $M_n(\mathbb{R})$ be the set of real $n\times n$ matrices. I've proved that the map $\left \|\cdot \right \| \mapsto \left \| A \right \| :=\sqrt{\text{tr}(A^tA)}$ is a norm. Then I defined the ...
0
votes
1answer
31 views

The continuity of infimum of a function

Let $(X,d)$ be a connected metric space and $(Y,d')$ is a compact metric space. Let $f$ be a continuous function from $(X\times Y,\max(d,d'))$ into $\mathbb{R}$. Because $Y$ is compact we can define: ...
0
votes
1answer
38 views

Computing the Manhattan Distance between two clusters of points. [closed]

We have two clusters of points: c1: (1, 1), (1, 2), (1, 3) c2: (2, 7), (2, 8), (2, 9) I know the Manhattan Distance formula is as follows: $d(a,b) = \sum|b_i - ...
5
votes
1answer
27 views

Componentwise Convergence in $\mathbb R^n$

I came across the following question while preparing an exercise for basic analysis: Suppose $d$ is some arbitrary metric on $\mathbb R^N$ and $(x^n) \subset \mathbb R^N$ converges to $x\in \mathbb ...
0
votes
1answer
27 views

Clarification about completeness of metric spaces

This is probably a very silly question but it bothers me for some time. We define a metric space $X$ to be complete if every Cauchy sequence in $X$ converges to some point in $X$. But any metric ...
3
votes
0answers
37 views

$\alpha$ exists so that for any points $x_n$ there is a point at average distance $\alpha$ from the $x_n$.

Let $X$ be a connected and compact metric space. Prove a real number $\alpha$ exists so that for every finite set of points $x_1,x_2,\dots, x_n\in X$ (not necessarily distinct) there exists $x\in X$ ...
4
votes
0answers
55 views

Theorem 3.7 in Baby Rudin: The subsequential limits of a sequence in a metric space form a closed set

Here's Theorem 3.7 in the book Principles of Mathematical Analysis by Walter Rudin, 3rd edition. The subsequential limits of a sequence $(p_n)$ in a metric space $X$ form a closed subset of $X$. ...
2
votes
1answer
27 views

Decide whether D is a distance function or not

Let $A$ be a set, $X:=\{x_1,...x_k\}$,$Y:=\{y_1,...,y_{k}\}$ $\subset \frak{P}$$(A)\setminus \emptyset$ subsets of the power set of $A$, both with cardinality $k$ and $d$ be a metric on ...
0
votes
0answers
20 views

$d:\mathbb N \times \mathbb N \to \mathbb R \ d(m,n)=0:m=n ; d(m,n)=1-\frac{1}{m+n}:m \neq n. $ Question convergence of sequences $(x_n)(y_n)(z_n)$

$d:\mathbb N \times \mathbb N \to \mathbb R \ d(m,n)=0:m=n ; d(m,n)=1-\frac{1}{m+n}:m \neq n. $ Question convergence of sequences $(x_n)(y_n)(z_n)$ and prove that $B_n=\{m \in \mathbb N : d(m,n)\leq ...
1
vote
2answers
31 views

Definition of interior

An interior point is defined as the following in the Euclidean space. If $S$ is a subset of a Euclidean space, then $x$ is an interior point of $S$ if there exists an open ball centered at $x$ ...
0
votes
1answer
24 views

Adding an isolated point to a Borel space

I have a Borel space $S$, which is basically a Borel subset of a Polish space. I want to add an isolated point $\alpha$ to $S$. Let $\overline{S}=S\bigcup \{\alpha\}$. Can I say that $S$ is clopen in ...
3
votes
2answers
377 views

Can the real vector space of all real sequences be normed so that it is complete ?

Let $X$ be the vector space of all real sequences . Does there exist a norm on $X$ which makes it complete ?
0
votes
1answer
21 views

What is the closure of an open ball $B_X(\mathbf{a},r)$ in $X=\mathbb{R}^n$?

Suppose we have the open ball $B_{X}(\mathbf{a},r)$ and the closed ball $\bar{B}_{X}(\mathbf{a},r)$ of radius $r$ about $\mathbf{a}\in\mathbb{R}^n=X$ with the Euclidean metric $d_2$. What is the ...
0
votes
1answer
16 views

$X,Y$ be real NLS ; $T:X \to Y$ be a linear map such that $\ker T$ is closed ; then does $T$ have closed graph?

Let $X,Y$ be real normed linear spaces and $T:X \to Y$ be a linear map with closed kernel ; then does $T$ have closed graph ? What if we assume arleast one of $X,Y$ to be complete ?
1
vote
2answers
35 views

On the dimension of a real Normed Linear Space possessing a certain property

Let $X$ be a real NLS such that for every proper subspace $Y$ of $X$ , $\exists x \in X$ such that $||x||=1$ and $dist (x,Y)=1$ ; then is $X$ finite dimensional ?
1
vote
1answer
12 views

$Y$ is a ( closed) proper subspace of a real NLS $X$ such that $dist (x,Y)=1$ for some $x \in X$ with $||x||=1$ ; is $Y$ finite dimensional?

Let $Y$ be a finite dimensional proper subspace of a real NLS $X$ , we know that we can find $x\in X$ ( depending on $Y$) , such that $||x||=1$ and $dist (x,Y):=\{||x-y||:y\in Y\}=1$ . I would like to ...
0
votes
0answers
14 views

Is this proof about metric spaces and boundaries correct?

Prove that $\partial(A\times B)=(\partial A\times \overline{B}) \cup (\overline{A} \times \partial B)$ given that $E$ and $F$ are metric spaces and contain $A$ and $B$ respectively Knowing that ...
5
votes
2answers
438 views

Is a space metric on the positive real numbers not complete?

Say we have a metric space $(\mathbb{R}^+, d)$ where the distance function is $d(x,y) = |x - y| + | 1/x - 1/y |$ Then I argue that this metric space is not complete: If we look at the Cauchy ...
2
votes
5answers
79 views

Cover $(0, +\infty )$ by open sets

Cover $(0, +\infty)$ by open sets $U_\alpha$ such that for any $\epsilon > 0$ there are points $x, y \in (0, +\infty)$ with $|x-y|<\epsilon$, not both belonging to the same $U_\alpha$ The ...
1
vote
2answers
60 views

Are $\{0\},\{1\}$ clopen in $\{0,1\}$ with the Euclidean metric? [closed]

Are $\{0\},\{1\}$ clopen in $\{0,1\}$ with the Euclidean metric? I think they are, but I would like a confirmation. Thank you.
0
votes
2answers
21 views

$X$ be finite dimensional real NLS , let $x \in X$ , does there exist $T \in \mathcal B(X)$ such that $\{T^n(x):n \in \mathbb N\}$ is dense in $X$?

Let $X$ be a finite dimensional real normed linear space , let $x \in X$ , then does there exist a continuous linear transformation $T:X \to X$ such that $\{T^n(x):n \in \mathbb N\}$ is dense in $X$ ? ...
0
votes
0answers
37 views

the point set is nowhere dense in $X$

I am trying to show for a metric space $X$, a set $\{x\}$ consisting of a single point is nowhere dense. I have proven it by showing $[{\{x\}}]^o = \emptyset$ where $[A]$ is the closure of the set $A$ ...
0
votes
0answers
36 views

Canonical metric on the suspension of a metric space

Let $(X,d)$ be a metric space. Is there a metric $d'$ on the (unreduced) suspension $\Sigma X = (X\times[-1,1])/\sim$ of $X$ such that $d'$ restricts to $d$ on $X\times \{1/2\}$? Further, we would ...
1
vote
0answers
49 views

Is it possible to prove this using set theory only and no more?

Got to prove: If $E$ and $F$ are connected sets, and $A$,$B$ are subsets of $E$ and $F$ respectively (but neither $A$ or $B$ are empty or fill $E$ and $F$entirely). Then $(A\times B)^c$ is connected ...
0
votes
1answer
12 views

Let $X$ be a non-zero real NLS , $x,y \in X$ , $B(x,r)\subseteq B(y,s)$ , then is it true that $r \le s$ ?

Let $X$ be a non-zero real NLS , $x,y \in X$ , $B(x,r)\subseteq B(y,s)$ , then is it true that $r \le s$ ? If $y=x$ then it is easy to see that that's the case . So I thought let $y \ne x$ ; I tried ...
1
vote
2answers
35 views

Fix some point $a \in M$. Prove that the function $f:M \to \mathbb R$ defined by $f(x) = d(a,x)$ is a continuous function on $M$

Let $M$ be a metric space with metric distance function $d(x, y)$, for $x, y \in M$. Assume that $M$ has only a countable or finite number of points, and assume that $M$ is connected. Fix some point ...
1
vote
2answers
35 views

Quotient of compact metric space is metrizable (when Hausdorff)?

Apparently it's a standard result that, although not every (Hausdorff) quotient of a metric space is metrizable, it always is metrizable when the space being quotiented is compact. Alas, I can't find ...
0
votes
1answer
27 views

Which of the options makes sense for this “boundary of a set” excercise

I've got a homework to do, I'm not sure if I'm missing parenthesis or something, I've got to prove this: For the metric spaces $E=(E,d_1)$ and $F=(F,d_2)$ with $A\subset E$ and $B\subset F$. ...
0
votes
1answer
45 views

Are proper maps compact?

Recently I learned about the notion of a proper map in metric spaces. Namely, if $X$, $Y$ are metric spaces, then a map $f:X\rightarrow Y$ is called proper iff for every compact set $K\subseteq Y$ the ...
0
votes
2answers
27 views

Calculate $p$-adic metric

Let $p$ be a prime. Define the $p$-adic modulus of $x$ on $\mathbb{Q}$ as $$ x= \frac{a}{b} \cdot p^{n}.$$ where $a$ and $b$ are relatively prime and do not contain $p$ as a factor as $|x|_p=p^n$. For ...
3
votes
2answers
33 views

Topology/ Metric on possibly unbounded functions

I am trying to think of a topology (possibly metric, as I am more used to think about things in metric spaces) on possibly unbounded functions (on $\mathbb{R}$) such that 1) convergence in that ...
1
vote
1answer
26 views

$d(a,X) = d(a,\overline{X})$ (distance from point to set is distance from point to closure)

I'm trying to understand this proof that: $$d(a,X) = d(a,\overline{X})$$ The proof says: Since $X\subset \overline{X}$, then $d(a,\overline{X})\leq d(a,X)$. We just need to show that the $<$ ...