Linked Questions

4
votes
1answer
941 views

Writing $(a,b)$ as a disjoint union of closed intervals [duplicate]

I've been thinking about the following question: Is it possible to write $(a,b)$ as a disjoint union of closed intervals? My first guess was no, but then I figured the question might be hiding ...
0
votes
1answer
437 views

Can the closed interval [0, 1] be expressed as the union of a sequence of disjoint closed subintervals each of length smaller than 1? [duplicate]

Can the closed interval [0, 1] be expressed as the union of a sequence of disjoint closed subintervals each of length smaller than 1? Explain. I have been trying to figure out this problem, and I do ...
2
votes
1answer
261 views

Prove that there does not exist a sequence $I_n=[a_n,b_n]$ such that $\bigcup_{n}I_n=[0,1]$ [duplicate]

Prove that there does not exist a sequence $I_n=[a_n,b_n], n=1,2,\ldots $ of nonempty, pairwise disjoint intervals such that $\cup_{n}I_n=[0,1]$. My solution -- is it correct? My idea is that taking ...
0
votes
0answers
213 views

Why can't a closed interval on R be written as the disjoint union of countably infinite many closed intervals? [duplicate]

As stated in the title: Why can't a closed interval $[a,b]$ on R be written as the disjoint union of countably infinite many closed intervals $[a_i,b_i]$?
3
votes
2answers
2k views

Is $[0,1]$ a union of family of disjoint closed intervals?

According to this question, $[0,1]$ cannot be written as union of countable disjoint closed sets, is the same true about (uncountable) family of disjoint closed intervals ?
4
votes
1answer
5k views

Can closed sets in real line be written as a union of disjoint closed intervals?

It is known that open sets in real line can be written as a countable union of disjoint open intervals. (link) I'm curious that if there is similar statements for closed sets in real line.
2
votes
2answers
631 views

Is Every (Non-Trivial) Path Connected Space Uncountable?

I know that every non-trivial metric space with more than one point which is connected is uncountable. However, if we don't demand that the space be a metric space, we can find examples of such odd ...
2
votes
1answer
2k views

Prove that the Cantor set cannot be expressed as the union of a countable collection of closed intervals

Prove that the Cantor set cannot be expressed as the union of a countable collection of closed intervals whereas it's complement can be expressed as the union of a countable collection of open ...
2
votes
2answers
2k views

The interval $[0,1]$ is not the disjoint countable union of closed intervals.

The following proof was suggested: suppose [0,1] was the disjoint countable union of closed intervals. Write the intervals as $[a_n,b_n]$. Start by showing the set of endpoints $a_n, b_n$ is closed. ...
2
votes
2answers
2k views

Show that any open set $U\subseteq\mathbb{R}^n$ is a disjoint union of countably many intervals.

As the title says, I have to show that any open set $U\subseteq\mathbb{R}^n$ is a disjoint union of countable many intervals. Hello, my idea is the following: Consider any $x=(x_1,...,x_n)\in U\...
1
vote
1answer
295 views

Prove $\mathbb{R}$ is not a countable, disjoint union of proper closed subsets. How about $\mathbb{R}^n$?

The formulation of the problem states: Show that $\mathbb{R}$ is not a countable, disjoint union of proper, closed sets. Can this be generalized to $\mathbb{R}^n$? There are many questions very ...
1
vote
1answer
1k views

In $\mathbb R^p$:Every open subset is the union of a countable collection of closed sets & every open set is the countable union of disjoint open sets

Prove/Disprove that : $(i)$ Every open Set in $\mathbb R^p$ can be written as the union of countable number of disjoint open Sets. $(ii)$ Every open subset of $\mathbb R^p$ is the union of a ...
0
votes
2answers
335 views

Path-Connectedness in Uncountable Finite Complement Space

As the title said, I want to show that an uncountable finite complement space is path-connected. I found that this question is answered in here. However, I'm having trouble understanding its proof. I'...
0
votes
3answers
155 views

Continuous and uncountable function [closed]

Let $f:[a,b] \to \mathbb{R}$ be a non-constant continuous function. Show that $f ([a,b])$ is uncountable.
1
vote
1answer
56 views

Is it possible that there is a connected topological space without path-connected subspace?

Is it possible that there is a connected topological space without path-connected subspace? Furtherly ~ Is that any connected topological space $X$ always has dense path-connected subspace? Or Is ...

15 30 50 per page