Tagged Questions

3
votes
1answer
83 views

Is every discrete topological space orderable?

I apologize for asking a question in topological terms when it's not really about topology, but here goes: If $S$ is a set, is it always possible to totally order $S$ so that every non-minimal ...
1
vote
3answers
78 views

Question regarding well-ordering theorem

The well-ordering theorem states that every set can be well-ordered. That is, there is a total order on the set in which every non-empty subset has a least element. I was wondering whether it's also ...
2
votes
1answer
128 views

Is the set $S$ of functions $S = \{ f : \mathbb R \to \mathbb R \}$ from the reals to the reals a totally ordered set?

Is the set $S = \{ f : \mathbb R \to \mathbb R \}$ of functions $f$ from the reals to the reals a totally ordered set ? I think the question is clear. Note that there is a bijection to $g(x)$ gives ...
4
votes
3answers
296 views

How to define a well-order on $\mathbb R$?

I would like to define a well-order on $\mathbb R$. My first thought was, of course, to use $\leq$. Unfortunately, the result isn't well-founded, since $(-\infty,0)$ is an example of a subset that ...
5
votes
2answers
215 views

What does a well ordering of $\mathbb{R}$ look like? [duplicate]

Possible Duplicate: Is there a known well ordering of the reals? I am having a hard time wrapping my head around what a well-ordering of $\mathbb{R}$ looks like. I have seen the ...
2
votes
1answer
160 views

Is a chain-complete lattice a complete lattice without the axiom of choice?

This question is inspired by this question. Consider the following result: Let $(L,\leq)$ be a chain-complete lattice. Then $(L,\leq)$ is a complete lattice. Can this result be proven without ...
2
votes
2answers
153 views

Bourbaki-Witt fixed point theorem: two questions

Consider the following theorem: Let $f\colon E \to E$ have the propery that $f(x)\geq x$, where $(E,\leq)$ is a non-void partially ordered set with the property that every totally ordered subset of ...
8
votes
4answers
197 views

Is there an element with no fixed point and of infinite order in $\operatorname{Sym}(X)$ for $X$ infinite?

Let $X$ be an infinite set. Let $\operatorname {Sym}(X)$ denote the group of all bijections from $X$ onto itself. I have been thinking about the existence of elements of infinite order in this group. ...
4
votes
4answers
385 views

Prove that for any infinite poset there is an infinite subset which is either linearly ordered or antichain.

Let $(X, \leq)$ be an infinite poset. Prove that there is an infinite $X' \subset X$ for which holds one of the following: 1) Induced order on $X'$ is linear. 2) $(X', ...
2
votes
1answer
117 views

How different are inductively ordered set from posets that satisfy the ascending chain condition?

I'd like to show that posets that satisfy the ascending chain condition (ACC) have maximal elements. I noticed this statement looks much like Zorn's lemma which holds for inductively ordered sets. ...
1
vote
2answers
125 views

How does this statement on posets follow from Zorn's lemma?

Let $P$ be a poset with a maximal element $r$, i.e. an element $r\in P$ such that $r\geq x$ for all $x\in P$. One can think of the poset as a tree with root $r$. I want to show that there exists a ...