1
vote
3answers
39 views

For a set $\mathbb{X}$ given order relation extended from that of $\mathbb{R}$, if $\mathbb{X} \supseteq \mathbb{R}$ then $\mathbb{X} = \mathbb{R}$?

For a set $\mathbb{X}$ given order relation extended from that of $\mathbb{R}$, if $\mathbb{X} \supseteq \mathbb{R}$ then $\mathbb{X} = \mathbb{R}$ ? Motivation for this question rose from an ...
1
vote
0answers
29 views

Order-preserving embeddings

(Follow-up to Existence of a utility function on the reals.) Say we have a totally ordered set $X$ which has a countable, dense subset $C$. I believe we can find an $f:C\to\mathbb R$ which is ...
2
votes
1answer
54 views

How to rigorously prove that these two sets have different order types?

Let $A$ and $B$ be two given ordered sets with the linear (or total) order relations $<_A$ and $<_B$, respectively. Then $(A,<_A)$ and $(B,<_B)$ are said to be of the same type if there ...
0
votes
1answer
52 views

Question about posets and maxima/minima

A thought just occurred to me, thinking about posets and maxima/minima... This is a "little" question just to make sure I am really grasping the definitions here: if $E$ is partially ordered by a ...
1
vote
1answer
100 views

Total Ordering and relation

Consider the relation $<$ on $\mathbb{Q}$ defined by: $(m, n) < (j, k) \iff jn-mk \in \mathbb{N}.$ Where $m, j \in \mathbb{N}$ and $n, k\in\mathbb{Z}$ I want to show that $<$ is a total ...
3
votes
4answers
130 views

$\sup(S)$ does not belong to $S$?

Problem: can anyone come up with an ordering of $\mathbb{N}$ different than the standard one we know, where we can find a subset $S\subset \mathbb{N}$ in a way such that $\sup(S)$ exists in ...
2
votes
3answers
210 views

Why is 'Antisymmetry' named so?

So when we talk about order relations for the familiar number systems, we are always introduced to the antisymmetry property which is $x \le y, x \ge y \implies x=y$. When I think of the word ...
1
vote
1answer
379 views

Rudin Theorem 1.11

After spending a few hours trying to understand Theorem 1.11 in Rudin's Principles of Mathematical Analysis, I still don't follow the proof. 1.11 Theorem Suppose $S$ is an ordered set with the ...
1
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1answer
3k views

Greatest lower bound property and least upper bound property

A Completeness principle in mathematical analysis is a principle by the help of which we can establish (prove) the completeness of an ordered field( About whom I am going to post one question later). ...
3
votes
1answer
74 views

For some sets $S\subseteq T\subseteq U$, when is $\inf_T S=\inf_U S$?

I've been struggling with the following problem I found for a while now: Suppose $(T,\preceq)$ is a partially ordered subset of $(U,\preceq)$ and $S\subseteq T$. If $\inf_T S$ and $u=\inf_U S$ both ...