All Questions
Tagged with well-orders general-topology
30
questions
2
votes
1
answer
169
views
When does a set of real numbers have a minimal element?
I Believe the answer is when the set is closed in the sense of standard topology (we exclude $\mathbb{R}$) itself.
Examples:
Point sets have a minimum, and they are closed.
Closed interval also have ...
1
vote
1
answer
78
views
Complete iff Compact in Well-Ordered space
Let $T=(S, \leq, \tau)$ a well-ordered set equipped with order topology (defined here).
Definition 1: $T$ is called complete iff every non-empty subset of $T$ has a greatest lower bound (inferior) and ...
- 188
0
votes
0
answers
37
views
The arithmetic of first uncountable ordinal number
I think, I know the proof of 1+ω0 = ω0. (ω0 is countable ordinal s.t ω0=[N]). To prove this, I can define a function f: {-1,0,1,2,...}->{0,1,2,...} by f(x)=x+1. But if ω1 is first uncountable ...
2
votes
1
answer
48
views
Intersection of a chain of closed sets
I was recently attempting proving a conjecture of mine about the existence of certain minimal nonempty closed sets in a topological space. I opted to use Zorn's Lemma, and the proof would go through ...
- 1,249
3
votes
2
answers
324
views
Product of Countable Well-Ordered Set with $[0,1)$ is Homeomorphic to $[0,1)$
As part of a proof that the long line is locally Euclidean, I'd like to prove the following:
Proposition. If $A$ is a countable well-ordered set, then $A \times [0,1)$ with the dictionary order is ...
- 2,221
0
votes
0
answers
170
views
Cantor-Bendixson derivative sets
I try to show that a compact subset $A\subset \mathbb{C}$ is at most countable if and only if there exists a countable ordinal number $\alpha$ (i.e $\alpha <\omega_{1},$ where $\omega_{1}$ is ...
- 51
1
vote
0
answers
60
views
Convex subspace of linear continuum is connected [Theorem 24.1, Munkres]
My questions from the same excerpt quoted here,
Proof. $\ \ $ Recall that a subspace $Y$ of $L$ is said to be convex if for every pair of points $a,b$ of $Y$ with $a<b$, the entire interval $[a,b]$...
1
vote
0
answers
46
views
Discussing Ex 13. Section 3. p-29, Munkres' Topology 2E. [duplicate]
In Exercise 13, section 3 on page 29 of Munkres’ Topology 2E, the problem is stated as follows.
Prove the following:
Theorem. If an order set $A$ has the least upper bound property, then it has the ...
- 307
1
vote
1
answer
63
views
Open sets in $S_\Omega$ (minimal uncountable well ordered set) with the order topology.
$X$ is a well ordered set. $S_\Omega= \{ x:x\in X\text{ and } x<\Omega\}$ such that it is the minimal uncountable well ordered set. The section $S_\Omega$ of $X$ is uncountable and any other ...
- 2,117
2
votes
1
answer
78
views
If $Y$ is a closed and bounded subset of an ordered space $X$ then is it compact?
Let be $(X,\preccurlyeq)$ a totally ordered set. So if $Y$ is a bounded and closed subset of $X$ with respect the order topology then is it compact? if this is not generally true then is it true when $...
- 4,906
0
votes
1
answer
189
views
Zorn's lemma and maximal elements
This continues a discussion begun at checking Zorn's lemma on an example where an example was offered to help understand maximal elements and Zorn's lemma. That example used the set {1,...,100} ...
- 23
0
votes
1
answer
256
views
First countability requirement of the Sequence Lemma
Consider the Sequence Lemma:
Let $X$ be a topological space, $A\subseteq X$ any subset and $x\in X$. If there is a sequence of points in $A$ converging to $x$, then $x\in\bar A$; the converse holds ...
- 313
0
votes
1
answer
42
views
Is the proper segment of a TOSET is initial segment? [closed]
I can't fimd its answer.
I have learnt that for a well ordered set proper segment is the initial segment. But i am unable to find segment of a toset which is proper and not initial.
0
votes
2
answers
43
views
Order topology related question
Say I have two topological spaces $X,Y$ s.t $Y$ has the order topology, and $f,g$ are two continuous functions $g,f:X\rightarrow Y$. I want to show that the set $O=\{x:f(x)> g(x)\}$ is open in $X$. ...
- 4,653
0
votes
0
answers
120
views
On an exercise concerning well-ordering
Working on set theory I stumbled across this exercise:
Let $\{A_\alpha: \alpha \in \mathcal{A}\}$ be a family of well-ordered sets. If for every pair of sets, one is an ideal of the other, prove that ...
- 175