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)

3
votes
2answers
165 views

Why is a rectangle not a neighborhood of its corners?

I'm trying to puzzle out a statement given in the Wikipedia article on topological neighborhoods, which uses this definition: If $X$ is a topological space and $p$ is a point in $X$, a ...
0
votes
2answers
155 views

Finding an isomorphism between $(U\otimes V)^{\mathsf T}$ and $B(U \times V,\mathbb{R})$

Prove the isomorphism between $(U\otimes V)^{\mathsf T}$ and $B(U \times V,\mathbb{R})$, where $B$ is the collection of all bi-linear mappings. In order to do so, give a natural isomorphism between ...
2
votes
1answer
155 views

How do I sketch the following metrics:

In $\mathbb{R}^2$ sketch $B((1,2),3)$, the open ball of radius $3$ at the point $(1,2)$, with the following metrics: a.) the post-office metric given by $$d(x,y) = \left\{ \begin{array}{l l} ...
2
votes
3answers
85 views

Does a norm have to map to $\mathbb R$?

Wikipedia defines a norm on a vector space $V$ as a function $p : V \mapsto \mathbb R$. I've seen this defined similarly elsewhere. However, it seems to me that a real codomain isn't always necessary. ...
3
votes
3answers
3k views

Show that $d_b(x,y)=\frac{d(x,y)}{1+d(x,y)}$ is a metric. [duplicate]

where $(X,d)$ is a metric and $x,y \in X$. I know we need to show: non-negativity: $d(x,y)\geq$ 0 $d(x,y)=0$ if and only if $x=y$ symmetry: $d(x,y)=d(y,x)$ $d(x,z)\leq d(x,y) + d(y,z)$ I think we ...
4
votes
5answers
520 views

How to show that a point is not an interior point?

I understand that in order to show that a point, $x$, is an interior point of some set $A \subset B$, where $(B,d)$ is a metric space you just need to show that you can have an open ball around $x$ ...
2
votes
2answers
180 views

If $f_n(x)=x^n$ converges to $f$, why is $f$ not continuous?

I was reading my Analysis course notes and had some trouble. I hope you can help me. Let $C(X)=\{ f | f:X \longrightarrow \mathbb{R} \text{ is a continuous function}\}$. It was already stated and ...
4
votes
2answers
667 views

Metric Spaces Analysis

Let $(X,d)$ be a metric space and for $x,y \in X$ define $d_b(x,y) =$ $ \dfrac{d(x,y)}{1 + d(x,y)}$ a) show that $d_b$ is a metric on $X$ Hint: consider the derivative of $f(t)$ = $\dfrac{t}{1+t}$ ...
1
vote
1answer
105 views

Metric induced Topology

The Problem: Given a metric space $(X,d)$, define a new metric $d'$ on X by $$ d'(x,y)=\frac{d(x,y)}{d(x,y)+1} $$ Is the topology induced by $d'$ the same as the topology induced by d? Prove or ...
1
vote
2answers
142 views

Closure in Metric Space

I need help understanding Theorem 2.27(c) in Rudin. If $X$ is a metric space and $E \subset X$, and if $E'$ denotes the set of all limit points of $E$ in $X$, then the closure of $E$ is the set $\bar ...
1
vote
1answer
44 views

Isometric involutions and sections

I have a metric space $X$ and an isometric involution defined on it $i:X\rightarrow X$. My intuiton tells me that I can find a (continous) section $s:X/i \rightarrow X$. Is this true? Any references ...
2
votes
1answer
383 views

Prove that two normed linear spaces are equivalent as metric spaces if and only if the norms are equivalent?

We have the two norms $\|\cdot\|_a$ and $\|\cdot\|_b$ on the vectorspace V. They're equivalent if there exists a $k>0$ and $K>0$ so that $k\|\cdot\|_a\le\|\cdot\|_b\le$ K$\|\cdot\|_a$ for all ...
1
vote
1answer
468 views

Show that a map $f:X\to Y$ is onto iff $f(f^{-1}(C))=C$ for all subsets $C\subseteq Y$.

Show that a map $f:X\to Y$ is onto iff $f(f^{-1}(C))=C$ for all subsets $C\subseteq Y$. My workings so far: Because this is an if and only if proof we need to show it both ways. First let's assume ...
2
votes
2answers
61 views

Proving inequality with metrics

I was trying show that $|D(x,B) - D(y,B)| \le d(x,y)$ with $D(x,B) = \inf_{b \in B} d(x,b)$ and $(X,d)$ is a metric space. My try: $d(x,y) \ge d(x,b) -d(y,b) \ge \inf_{b\in B}d(x,B) - d(y,b)$ forall ...
1
vote
1answer
33 views

$\sup_{i \in I} \operatorname{dist}(x,S_i) \leq \operatorname{dist}(x, \bigcap_{i \in I}S_i)$

Prove that $$\sup_{i \in I} \operatorname{dist}(x,S_i) \leq \operatorname{dist}(x, \bigcap_{i \in I}S_i)$$ where $(X,d)$ is a metric space, $S_i \subseteq X : i \in I$, and $x \in X$. The question ...
2
votes
2answers
116 views

Why are the interior points in this question not the same?

I'm working on a question that wants me to write down the interior points of an interval contained in a metric space. $Let X=((1,7],d_{E})$ be a subspace of the metric space $(\mathbb{R},d_{E})$. Let ...
0
votes
2answers
420 views

Distance between two sets in a metric space in different conditions

let $(X,d)$ be a metric space and let $A,B\subseteq X$. we define the distance between $A$ and $B$ as: $$\operatorname{dist}(A,B)=\inf\{d(a,b):a \in A,b \in B\}$$ 1 show that for any $x \in X$, we ...
0
votes
2answers
67 views

Does this proof make sense and correct — is it written well enough?

I'm working on a tutorial question. The question asks whether the following claim is true or false, if it is true: one is supposed to provide a proof or counter-example otherwise if it's false. Let ...
6
votes
2answers
117 views

A question on Hamming metric/distance

Suppose $\sf{X}=\{0,1\}$, and $\sf{X}^n$ is the set of all binary sequences of length $n$. So the first question is that what does it mean by the convex closure of a subset $\sf{A}$ of $\{0,1\}^n$, ...
1
vote
1answer
77 views

Intersection of a Perfect and an open subsets of X

In Prof. George Bergman's Real analysis supplementary exercises, question 2.2:10. It is required to proof that the closure of the intersection of a perfect set $E$ with an open set $A$ is again ...
1
vote
1answer
193 views

A question on quasi-components

I have been doing some reading on general topology, connectedness in particular. Here is a question on a topological concept called quasi-component. Here is a definition: ...
6
votes
2answers
120 views

non compact nested closed sets in metric spaces.

Do you have an example for closed sets $...\subseteq F_4\subseteq F_3\subseteq F_2\subseteq F_1$ such that: $$\bigcap_{n=1}^\infty F_n=\emptyset $$ in $\mathbb{R}^n$ or a metric space?
9
votes
1answer
173 views

A natural-looking distance formula

The distance formula in one dimension is $$D_1 = |x_2-x_1|$$ From the Pythagorean theorem, the distance formula in two dimensions is $$D_2 = \sqrt{|x_2-x_1|^2 + |y_2-y_1|^2}$$ Now, in three ...
2
votes
2answers
67 views

Confused by an argument which is used in most triangle inequality proofs in metric spaces

I'm confused by the a proof of the triangle inequality. I was supposed to prove that a function is a metric, I proved everything else except the triangle inequality. Define $B(\mathbb{R})$ as the set ...
1
vote
5answers
151 views

A statement true about compacts but false about closed sets

Suppose that you have a metric space $X$. Could you give an example of a theorem or a statement that holds for compact sets but does not hold for closed sets? This question is motivated from a ...
2
votes
1answer
71 views

Is this proof correct and written in a understandable fashion?

Given a function $f(x,y) = \sqrt{(x_{1}-y_{1})^2 + (x_{2}-y_{2})^2 +...+(x_{n}-y_{n})^2}$ where $x,y$ $\in$ $\unicode{x211D}^n$, $f:\unicode{x211D}^n \times \unicode{x211D}^n \rightarrow ...
5
votes
1answer
255 views

$X$ metric separable then $C(X)$ separable

Is it true, that if $X$ is a separable metric space, then the space of all continuous functions on $X$ with the supremum metric is also separable?
2
votes
1answer
101 views

Clarification on the definition of the Lebesgue number of a metric space

Definition: Let $X$ be a metric space and $\mathcal{O}$ an open cover of $X$. A Lebesgue number for $\mathcal{O}$ is a positive number $\varepsilon$ with the property that every subset of $X$ of ...
5
votes
1answer
816 views

Is $\mathbf{R}^\omega$ in the uniform topology connected?

Let $\mathbf{R}^\omega$ be the set of all (infinite) sequences of real numbers. Then is this space connected in the uniform topology? How to determine this? The uniform metric $p \colon ...
0
votes
2answers
58 views

How to determine if this map is open or closed?

Given two supspaces $X:= [0,1]\cup[2,3]$ and $Y:=[0,2]$ of $\mathbf{R}$, let $f \colon X \to Y$ be defined as follows: $$f(x):= \left\{ \begin{array} {ll} x & \mbox{if $0\leq x\leq 1;$} \\ x-1 ...
2
votes
3answers
129 views

$d\left(\left(x_1,x_2\right),\left(y_1,y_2\right)\right)=|x_1-x_2|+|x_1-y_1|+|y_1-y_2|$ : complete?

Define the $\Bbb R^2$ metric $$ d\left( x,y \right) = \begin{cases} \left|x_2-y_2\right| &, x_1 = y_1 &&\text{(d1)}\\ \left|x_1-x_2\right|+\left|x_1-y_1\right|+\left|y_1-y_2\right| &, ...
1
vote
1answer
100 views

About Convergence of the Image of a Convergent Sequence Under a Uniformaly Convergent Sequence of Functions

Let $X$ be a topological space and $Y$ a metric space. Let $f_n \colon X \to Y$ be a sequence of continuous functions. Let $x_n$ be a sequence of points of $X$ converging to a point $x \in X$. Suppose ...
4
votes
3answers
317 views

Equivalent characterization of chain connectedness of a metric space

I'm having difficulty with proof. It is that the following is an equivalent characterization of chain connectedness for a metric space $M$: Point-wise boundedness at a point of an equicontinuous ...
1
vote
3answers
485 views

Infimum of distance in compact metric spaces.

Let $(A,d)$ be a metric space with $B\subseteq A$. If $B$ is compact, then it is bounded and closed. If $y\in A$ then there exists $x\in B$ so that $\inf\{d(y,z) : z\in B\} = d(y,x)$. It is reasonable ...
7
votes
2answers
306 views

Delta in continuity

Let $f: [a,b]\to\mathbb{R}$ be continuous, prove that it is uniform continuous. I know using compactness it is almost one liner, but I want to prove it without using compactness. However, I can use ...
1
vote
1answer
66 views

Sequences of functions

Suppose that $X$ is a compact metric space. Let: (a) $(f_n)$ be a sequence of real-valued continuous functions on $X$ (b) $(f_n)$ converges pointwise to a continuous function $f$ on $X$ (c) $f_n(x) ...
0
votes
1answer
213 views

What about the continuity of these functions in the uniform topology?

Let $f$, $g$, $h \colon \mathbf{R} \to \mathbf{R}^\omega$ be defined by $$\begin{align*} f(t)&:=(t,2t,3t,\ldots),\\\\ g(t)&:=(t,t,t,\ldots),\\\\ ...
1
vote
2answers
385 views

Totally disconnected metric space

I want to prove that a certain metric space is totally disconnected. In a metric space context this is the same as saying that every connected component is a singleton. I think another way of ...
2
votes
2answers
888 views

How to prove that the uniform topology is different from both the product and the box topology?

Let $J$ be an arbitrary index set. Then how to prove that the uniform topology on the Cartesian product $\mathbf{R}^J$ of the set $\mathbf{R}$ of real numbers with itself is different from both the ...
5
votes
2answers
603 views

How is the metric topology the coarsest to make the metric function continuous?

Let $X$ be a metric space with metric $d$. If $\mathcal{T}$ is a topology on $X$ such that the function $d\colon X \times X \to \mathbb{R}$ is continuous, then how to show that $\mathcal{T}$ is finer ...
4
votes
2answers
324 views

Continuous linear functionals

Let L be a continuous linear functional on a metric linear space X. Prove: L(S) is a bounded set for any bounded subset S of X. The metric is translation invariant.
2
votes
3answers
490 views

Example of Homeomorphism Between Complete and Incomplete Metric Spaces [duplicate]

Is it possible to have a homeomorphism between a complete metric space and an incomplete one? If so, what examples can be given?
2
votes
0answers
393 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) := ...
0
votes
0answers
72 views

Set of continuous function defined on some segment $[0,a]$: completeness

Let $S$ be the set of continuous function $f$ defined on some segment $[0,a_f], a_f \ge 0$, and such that $f(0)=0$. For $f$ and $g$ in $S$, let $$ c_{fg}=\max\{z \in [0,a_f \land a_g] : f(x)=g(x) ...
3
votes
2answers
51 views

Is it true that, $A,B\subset X$ are completely seprated iff their closures are?

If $A,B\subset X$ and $\overline{A}, \overline{B}$ are completely seprated, so also are $A,B$. since $A\subset \overline{A}$, $B\subset \overline{B}$ then, $f(A)\subset f(\overline{A})=0$ and ...
2
votes
1answer
48 views

All metrics on set of size 2

I was trying to determine all metrics possible on a set $X$ when size of $X$ equals two. It is clear that the discrete metric is one possible metric. But is the only requirement that the metric assign ...
2
votes
1answer
268 views

Subspace $Y$ of metric space with finitely many points is complete.

Show that if a subspace $Y$ of a metric space consists of finitely many points, then $Y$ is complete. This is what I have so far, but I don't know where to go from here: Suppose the the subspace ...
1
vote
1answer
719 views

Show $\mathbb R^n$ is complete.

Show $\mathbb R^n$ is complete. At this point, I am trying to work through the problem in my textbook, there is one step that I do not understand and would like explained. Here's my proof so far: ...
1
vote
1answer
79 views

a counterexample for Uniform Spaces

Uniform Space is a generalization of metric spaces . In a uniform space the closure of a singleton $\{x\}$ is the intersection of all neighborhoods of $x$. Find an infinite topological space such ...
1
vote
2answers
473 views

Bounded functions on subsets of Euclidean space

It is known that given any closed and bounded $X \subseteq \mathbb{R}^n$ and a bounded continuous function $f : X \to \mathbb{R}$, $f(X)$ has a minimum value and maximum value. This can be proved by ...