0
votes
2answers
43 views

basic question of topology involving compactness and convexity

Consider in $R^n$ a compact and convex set $A$ with $int(A) $ nonempty. then $\overline{int(A)} = A$ ?. i have no idea to prove this. In this direction i only know the following (and hard to ...
0
votes
2answers
21 views

Is a closure a disjoint union of limit points and isolated points

Definition) A point $x\in X$ is a limit point of S if every ball $B(x;r)$ contains infinitely many points from $S$. A point $x\in X$ is called an isolated point of S if $\exists r > 0$ such that ...
8
votes
2answers
31 views

Product topology and standard euclidean topology over $\mathbb{R}^n$ are equivalent

I would like to know why the product topology and the standard euclidean topology over $\mathbb{R}^n$ are equivalent. I already found the question here: Showing that the product and metric topology ...
1
vote
1answer
53 views

Using the topology of uniform convergence for functions over non-compact spaces

Let $(X, d)$ be a (complete) metric space, and $C(X)$ be the space of continuous maps over $X$. If $X$ is compact, one often uses the topology of uniform convergence when analyzing $C(X)$. If $X$ is ...
0
votes
0answers
37 views

Onto continuous function on a compact metric space is isometry. [duplicate]

Let $K$ be a compact metric space with metric $d$ and suppose $f:K\rightarrow K$ is continuous and surjective (onto), and satisfies $d(f(x),f(y))\leq d(x,y),\,\forall x,y\in K$. How can we prove that ...
0
votes
1answer
28 views

A simple question on Hausdorff distance

Let $(A_n)$ be a sequence of compact sets in $R^n$ and consider $K$ and $A$ compact sets in $R^n$. Suppose that $A_n \cup K \rightarrow A \cup K$ in the Hausdorff distance. Then $$ A_n ...
2
votes
1answer
38 views

Using Cantor's intersection theorem

Assume $f: X \rightarrow X$ is a continuous map where X is a compact metric space. Prove that there exists a non-empty set $A \subset X$ such that $f(A) = A$. (Hint: Set $F_1 = f(X), F_{n+1} = ...
0
votes
1answer
29 views

Show that the interior of the set A is empty?

Consider $A = \{(x, \sin\frac{1}{x}) \mid 0< x \leq 1 \}$, a subset of $\mathbb R^2$. Find int($A$). We can see graphically that the interior of $A$ is definitely empty, but I want to check by the ...
1
vote
1answer
37 views

Proving the distance function $|f - g|_u = \sup \{ |f(x) - g(x)|: x \in S \}$ defines a metric space

Let $S$ be a closed and bounded subset of $\mathbb{R}$. Define the "functional distance" between $f$ and $g$, both functions from $S$ to $\mathbb{R}$, to be $$ |f - g|_u = \sup \{ |f(x) - g(x)|: x ...
3
votes
3answers
43 views

Closed and Connected Subset of a Metric Space

My English may not be perfect since I'm not a native speaker, so please do point out the grammar mistakes if there are any. I've been reading Conway's "Functions of One Complex Variable", and ...
2
votes
1answer
30 views

How much close should two spaces be in the Gromov-Hausdorff distance to be homeomorphic?

Here I am considering the Gromov-Hausdorff convergence for metric spaces. I know two compact metric spaces are isometric if and only if $d_{GH}(X,Y)=0$, where $d_{GH}$ denotes the Gromov-Hausdorff ...
-1
votes
1answer
27 views

proving the equivalence of to metrics

Any hints as to how I can prove that $(\mathbb{R}^n,d_\infty)$ and $(\mathbb{R}^n,d_T)$ are topologically equivalent. Where $$d_\infty = \sup\{|x_i-y_i|, |x_2-y_2|,..,|x_i-y_i|\} $$ and $$d_T = ...
0
votes
1answer
41 views

Proving a basis exists

How would I show that there exists some set of open balls with rational radius and rational centre such that they are a subset of the reals.That is, $\exists p,q\in \mathbb{Q} $ and $ r,x \in ...
3
votes
4answers
882 views

Example of two open balls such that the one with the smaller radius contains the one with the larger radius.

Example of two open balls such that the one with the smaller radius contains the one with the larger radius. I cannot find a metric space in which this is true. Looking for hints in the right ...
1
vote
3answers
53 views

proving topological equivalence

How would I show that $$d_E = \sqrt{\sum_{i=1}^n(x_i-y_i)^2}$$ and $$d_\infty= \sum_{i=1}^n|x_i-y_i|$$ and $$\sup\{|x_1-y_1|,|x_2-y_2|,...,|x_n-y_n|\}$$ are topologically equivalent on $\mathbb{R}^n$? ...
1
vote
2answers
73 views

A metric space is compact if and only if its complete and totally bounded.

I just need help with one direction. In particular suppose that $(X,d)$ is complete and totally bounded. Suppose that $X$ is infinite. Take any infinite subset $Y$ in $X$ and then just show that $Y$ ...
1
vote
2answers
45 views

proving a metric

I'm trying to show that given a metric $d(x,y)$ show that $d_0(x,y) =\frac{d(x,y)}{1+d(x,y)}$ is also a metric.. It's trivial to show the first two properties, that is, $d_0\geq 0 $ & for $x=y ...
0
votes
0answers
17 views

Hairy ball theorem, projections and L.I. vectors

I'm trying to understand this paper which proves that not every unimodular row is completable by invertible matrices: Why we have these implications: There are two linearly independent vectors at ...
2
votes
1answer
28 views

Suppose two metrics induce the same bornology; do they necessarily induce the same topological space?

Every metric space $(X,d)$ gives rise to a bornology by asserting that $A \subseteq X$ is bounded iff there is an open ball (of finite radius) that includes $A$. Now just because two metrics induce ...
1
vote
1answer
24 views

Give an example of a metric space $(X,d)$ and $A\subseteq X$ such that $\text{int}(\overline{A})\not\subseteq\overline{\text{int}(A)}$ and vice versa

Give an example of a metric space $(X,d)$ and $A\subseteq X$ such that $\text{int}(\overline{A})\not\subseteq\overline{\text{int}(A)}$ and ...
0
votes
0answers
25 views

Some fundamental relations in topology

Are the following relations correct? $\ \{ Normed\, Vector\, Spaces\} \subset \{Topological\, Vector\, Spaces\} \subset \{Uniform \,Spaces\} \subset \{Topological\, Spaces\}$ Then $\ \{Normed\, ...
1
vote
1answer
67 views

Metrics and Continuous Functions

The following question reads as i) Given an example of infinite metric spaces $(X,d)$ and $(Y,\delta)$ such that every function $f\colon X\to Y$ is continuous. ii) Is it possible to give an ...
0
votes
1answer
46 views

Are there path-connected but not polygonal-connected sets?

This question came up in my mind. In the scope of normed spaces, does there exist a path-connected but not polygonal-connected set? I'd rather say no for open sets (my intuition is that ...
0
votes
0answers
35 views

NonCountable Product of Discrete Space {0,1}

Is the product of $\{0,1\}^I$ for any $I$ metrizable (with the product topology)? It Would be helpful to see proof\disproof idea. Thank you
1
vote
1answer
21 views

Compact homeomorphic non bilipschitz homeomorphic metric spaces

Statement: Let $(X,d)$, $(Y,r)$ be homeomorphic compact metric spaces. Then, there exists a bilipschitz homeomorphism between the two. Problem: As I have no clue whether the statement is true or ...
3
votes
1answer
33 views

Pseudometrics and non-expansive maps

Let $X$ be any set, and $(Y,\rho)$ be a pseudometric space. For a collection of functions $F\subseteq Y^X$ define $$ d_{F,\rho}(x',x''):=\sup_{f\in F}\rho(f(x'),f(x'')). $$ It is easy to show that ...
1
vote
1answer
25 views

Extension of a metric from closed subspace of a metrizable space to the whole space

How to prove: Let M be a closed subspace of a metrizable space X. Then, any metric on the subspace M can be extended to a metric on X.
0
votes
2answers
31 views

Distance between a point and a set

The problem I'm trying to solve is Prove that $d(a, B \cup C)$ is the smaller of $d(a,B)$ and $d(a,C)$ for a point $a$ and subsets $B, C$ of a metric space. So I think what I need to show is ...
0
votes
2answers
47 views

A open subset of $\Bbb R$

Given the definitions in Open Subsets of open sets I need to prove that $\{x \in \Bbb R : |x|>2\}$ is open in $(\Bbb R , d_E)$ This seems to be true, however I don't know how to prove it without ...
3
votes
2answers
48 views

closed ball in euclidean space

In general metric spaces the closed ball is not the closure of an open ball. However, I read that in the Euclidean space with usual metric, closed ball is the closure of an open ball. I'm having ...
0
votes
1answer
55 views

The Solution to Exercise 4, page 12, of Gamelin's “Introduction to Topology”.

I'm looking at exercise 4 in page 12 of Gamelin's Introduction to Topology. The problem is stated as follows: Suppose that $F$ is a subset of the first category in a metric space $X$ and $E$ is ...
1
vote
1answer
41 views

corollary to baire category theorem

I'm studying topology with gamelin and greene's text and I came across a corollary to the baire category theorem which states that "Let (En) be a sequence of nowhere dense subsets of a complete ...
2
votes
3answers
70 views

show the supremum of the distance function of a compact metric space is finite

Let $X$ be a compact topological space and $(Y,d)$ be a metric space. Show that for every pair of continuous functions $f\colon X\to Y$ and $g\colon X\to Y$, the extended real number $$ ...
2
votes
0answers
44 views

Zer0-dimensional, countable, 1st countable T1 space is metrizable?

Show that every countable, first countable, zero-dimensional T1 space $X$ is metrizable. I know that T1 space means that all its singletons are closed. Also, zero-dimensional means that $X$ has a ...
1
vote
1answer
45 views

Continuous functions on a closed subset of a topological space

Let $X$ be a topological space with $Y$ a closed subspace with relative topology. If $f:Y \rightarrow Z$ is a continuous map of topological spaces, then can $f$ always be extended to be from $X$ to ...
1
vote
4answers
72 views

closed ball in general metric space?

Is it true that in general complete metric space $(M,d)$, a closed ball of radius $r$ centered at $p\in M$ is always compact? That is, the ball is the set of all points $\left\{x:d(x,p)\leq ...
0
votes
2answers
90 views

theorem compactness and Hausdorff

I have this theorem "$X$ is compact $\leftrightarrow\exp X$ is compact", but i can not find source of it. It concerns Hausdorff metric.
0
votes
2answers
63 views

Metric on the line (1-Dimensional space)

Is the Euclidean metric the only possible metric in real line? Recall that the Euclidean metric is given by $d(x,y) = |y-x|$, and if there is another metric, are these two equal? what about the metric ...
0
votes
1answer
43 views

Covering of closed unit ball with closed balls.

Notations and definitions Let $E$ be a finite dimensional vector space with norm $||\;||$. Let $B$ denote the closed unit ball in $E$ and $B_r[a]$ the closed ball centered at $a$ with radius $r$. ...
0
votes
2answers
83 views

How can the y-axis in $\mathbb{R^2}$ be open?

I have read that $\{(x, \frac{1}{x}): x \neq 0\}$ is closed in $\mathbb{R^2}$. So hence the complement of this set, $\{x = 0\}$, i.e. the y-axis must be open? But we cannot put an open ball with ...
0
votes
1answer
56 views

In what metric spaces is a closed and bounded set compact?

Is there a characterization of a metric space $X$ such that for every $A\subseteq X$, $A$ is compact iff $A$ is closed and bounded? Something that generalizes $\mathbb R^n$?
0
votes
1answer
41 views

Definition of a nowhere dense set

I'm currently studying metric spaces through Gamelin and Greene's Introduction to Topology. While studying about completeness I got stuck with this concept of nowhere dense subset. The book defines a ...
4
votes
2answers
73 views

Is $\mathbb R^{\omega}$ homeomorphic to $\mathbb R^{\omega} \times \mathbb R^{\omega}$?

As a study exercise, I'm trying to find a topological space $X$ which is homeomorphic to $X \times X$. I began thinking of simple examples involving $\mathbb R$ but then realized my best bet would be ...
2
votes
2answers
52 views

Is this a metric on R?

For $x,y \in \mathbb{R}$, define $d(x,y) = \sqrt{|x-y|}$. Is this a metric on $\mathbb{R}$? It's clear that $d(x,x) = 0$ and $d(x,y) = d(y,x)$ for all $x,y \in \mathbb{R}$. The triangle inequality ...
2
votes
1answer
30 views

prove the set of all spheres with rational center and radius is countable

Prove the set of all spheres in $\mathbb{R}^3$ with rational center and radius is countable. I have two ideas. Is either one better than the other? 1) let $(x,r)$ represent a sphere in ...
0
votes
0answers
28 views

Sequence Criterion of Continuity and Divergent sequences

A function $f : (X, d_X) \to (Y, d_Y)$ between metric spaces is continuous iff $$ \lim_{n \to \infty} f(x_n) = f(\lim_{n\to \infty} x_n) $$ for each convergent sequence $(x_n)$ in $(X, d_X)$. So if I ...
3
votes
3answers
71 views

Supposed X and Y are metric spaces and $f:X \rightarrow Y$ and $g:X \rightarrow Y$ are continuous prove $\{x \in X:f(x)=g(x)\} $ is a closed set

Supposed X and Y are metric spaces and $f:X \rightarrow Y$ and $g:X \rightarrow Y$ are continuous prove $\{x \in X:f(x)=g(x)\} $ is a closed set I don't have any idea where to start. Any suggestions? ...
2
votes
1answer
56 views

Prove that the union of the interior of a set and the boundary of the set is the closure of the set

I'll denote closure of A with $A_C$ because I cant get the bar for some reason. also $Int(A)$ is interior of A, $Bdry(A)$ is the boundary of A and A' the accumulation points. I'm trying to prove the ...
2
votes
1answer
98 views

$\ell^{\infty}(\mathbb N)$ is not a separable space

I have to prove that $\ell^{\infty}(\mathbb N)$ is not separable. My attempt Consider a SUBSET $V$ of $\ell^{\infty}(\mathbb N)$ consisting of bounded sequences that have only $0$, $1$ entries, e.g. ...
4
votes
3answers
40 views

How to prove that for any $n\in N$, there exists a subset of real line that has nonempty $(n-1)^{th}$ derived set but an empty $n^{th}$ derived set?

How can we prove that for any positive integer n, i.e., $n\in \mathbb{N}$, there exists a subset of real numbers, i.e., $E\subset \mathbb{R}$, that has nonempty $(n-1)^{th}$ derived set but an empty ...