# Tagged Questions

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### Well-ordered family of open sets.

Let $X$ be a second countable space. I $\textbf{A}$ is a family of open set well-ordered by inclusion prove that this family is numerable. I have the following idea: Let $\textbf{B}$ be a ...
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### Existence of certain uncountable closed sets in the order topology

This is a proof-verification request. Let $\Omega$ be the set of countable ordinals, $\omega_1$ the first uncountable ordinal, and $\Omega^*=\Omega\cup\{\omega_1\}$. Remarkable properties of these ...
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### How to make an order isomorphism

Two linear orders $A$ and $B$ have starting points $a_0$ and $b_0$, and have cofinalities $\omega_1$. Let $(a_\alpha )_{\alpha<\omega_1}$ and $(b_\alpha )_{\alpha<\omega_1}$ be cofinal ...
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### Compactness and existence of Pareto-efficient cake partitions

I am trying to understand a fundamental statement in the theory of cake-cutting. BACKGROUND: There is a certain "cake" $C$ (a subset of $R^n$). The cake is divided among two agents, 0 and 1. Each ...
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### Principal order filters on a POSET

I have another problem, but in this one I have no idea how to start. Let be $(X,\leq)$ a POSET with a first element and gifted with the topology $\{ (a,\rightarrow) : a \in X \}$ (principal order ...
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### Extracting subnets from nets.

Let $x_i:\Lambda_i\to X_i$ be a net in a topological spaces $X_i$ that converges to $a_i\in X_i$ where $i\in\{1,2\}$ Does there exist a poset $\Lambda$ and subnets of $x_i:\Lambda_i\to X_i$ of the ...
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### What conditions on a totally ordered set imply that the closed intervals are connected?

It is well known that $[a,b]_\mathbb{R}$ is connected, while $[a,b]_\mathbb{Q}$ is not. I'm wondering, what conditions on a totally ordered set $T$ imply that for all $a,b \in T$, it holds that ...
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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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### Is there anything wrong with this use of the axiom of choice?

I have a proof of the following theorem: Let A be a finite set and X be a perfectly normal topological space, and let $\{(A,\succsim_x): x\in X\}$ be a family of binary relations on $A$ satisfying ...
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### Contractibility of topological spaces associated to posets

Suppose $\mathcal{P}$ is a partially ordered set. To $\mathcal{P}$ we can associate a simplicial complex $K(\mathcal{P})$ whose $n$-simplices are the chains of length $n+1$ in $\mathcal{P}$. Since ...
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### Left-isolated points in totally ordered set

Let $X$ be a totally ordered set. I say that a point $x$ is "left-isolated" if there is some $\alpha < x$ such that there is only one point $y$ which satisfies $\alpha < y \leq x$, that is, ...
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### What is an order reversing bijection on $\Bbb R$?

What is an order reversing bijection on $\Bbb R$ and why must it be continuous? Thanks
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### Is convex set interval or ray? [duplicate]

Given an ordered set $X$, say a subset $Y$ of $X$ is convex in $X$ if for each pair of points $a<b$ of $Y$, the entire interval $(a,b)$ of points of $X$ lies in $Y$. An interval is of the form ...
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### Reference request - the topology generated by upward and downward closures of antichains.

By definition, the order topology on a totally ordered set $T$ is the coarsest topology such that every subset of $T$ that can be expressed as the upward or downward closure of a singleton is closed. ...
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### Show that $X$ is separable.

I'm working through Kunen's Set Theory and I'm not sure how to proceed on part of one exercise. Let $X$ be compact Hausdorff and $\mathbb{O}_X$ be the poset of nonempty open sets of $X$ ordered by ...
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### Definition of Dedekind cut as initial segment

the definition of Dedekind cut by initial segment is correct: "let $\preceq$ be a linear ordering of a set $A$, and $B \subsetneqq A$, $B \neq \emptyset$, $B$ is Dedekind cut of $A$ under $\preceq$ ...
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### Every preorder is a topological space

Let $(X,\leq)$ be a preorder, i.e. $\leq$ is a reflexive and transitive binary relation on $X$. I want to show that $\leq$ induces a topological structure on $X$. Hence I need to specify when a subset ...
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### Are there only 2 clopen sets on real plane?

How can I prove that the only open and closed sets on the real plane are empty set and real plane itself? Preferably by using order theory. Thanks.
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### Give an example of a simply ordered set without the least upper bound property.

In Theorem 27.1 in Topology by Munkres, he states "Let $X$ be a simply ordered set having the least upper bound property. In the order topology, each closed interval in $X$ is compact." (The LUB ...
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### A problem about compact order topological space

This problem is from terrytao.wordpress.com/books/analysis-ii,Errata to the second edition (hardcover),P.390,Exercise 13.5.8 .I have difficulty proving this,appreciate any help! Show that there ...
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### A new(?) partial order on the set of continuous maps

Let $X,Y$ be topological spaces. Define a partial order on $\hom(Y,X)$ as follows: $f \leq g$ if $f^{-1}(U) \subseteq g^{-1}(U)$ for all open subsets $U \subseteq X$. Equivalently, $f(y)$ is a ...
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### Two topologies generated by a partially ordered set - are they equal? Are they $T_0$?

Let $P$ denote a partially ordered set, and consider two topological spaces with carrier $P$. The topological space whose closed sets are generated by the upsets and downsets of $P$. The ...
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### Is the topology generated by the open intervals of a partially ordered set necessarily T_0?

Let $X$ denote a set and consider a collection $C \subseteq \mathcal{P}X$. Let $\mathcal{T}$ denote the set of all topologies $T$ on $X$ such that $C \subseteq T$. Then $\bigcap \mathcal{T}$ is a ...
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### Does consistency of a test imply continuity of a preorder?

Let $F$ and $G$ be two real-valued probability distributions, considered as cadlag functions with the uniform norm [not the Skorohod metric], and let $\mathbb{F}_n$ and $\mathbb{G}_m$ be empirical ...
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### Order topology and partial order

Let $(X,\leq)$ be an ordered space and define on $X$ the order topology form the subbasis $\{(x,y), (x,\to), (\leftarrow,x) : x,y \in X \}$. Can you read the fact that $\leq$ is a total order on the ...
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### The relation between order isomorphism and homeomorphism

Let $X$ be a set and let $<_1,<_2$ be order relations on $X$. Let $T_1,T_2$ be the topologies induced on $X$ respectively. If $(X,T_1)$ is homeomorphic to $(X,T_2)$, does that imply that ...
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### Totally ordered set with greater cardinality than the continuum

Does there exist a totally ordered set $S$ with cardinality greater than that of the real numbers? Sequences are continuous functions with domain $\mathbb{N}$ and paths are continuous functions with ...
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### preorders induced by continuous functions to the reals

Any function to a (total) order induces a (total) preorder on its domain. What can be said about the total preorders induced by a continuous function from a "nice" topological space (for instance, a ...
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### An example with a non-increasing function

Let $\mathfrak{A}$ is a poset. For $a, b \in \mathfrak{A}$ we will denote $a \curlyvee b$ if only if there is a non-least element $c$ such that $c \leqslant a \wedge c \leqslant b$. Let ...
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### Does an isomorphism induce an order isomorphism?

Let $\mathfrak{A}$ is a poset. For $a, b \in \mathfrak{A}$ we will denote $a \curlyvee b$ if only if there is a non-least element $c$ such that $c \leqslant a \wedge c \leqslant b$. Let ...
Is every homeomorphism between topological spaces an order isomorphism (for orders of inclusion $\subseteq$ of sets)?
Denote by $\Sigma$ the collection of all $(S, \succeq)$ wher $S \subset \mathbb{R}$ is compact and $\succeq$ is an arbitrary total order on $S$. Does there exist a function \$f: \mathbb{R} \to ...