# Tagged Questions

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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### $\sup(S)$ 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 ...
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### 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 ...
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### 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 ...
### 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 ...