3
votes
1answer
26 views

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 ...
-1
votes
1answer
47 views

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 ...
4
votes
1answer
83 views

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 ...
2
votes
1answer
54 views

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 ...
1
vote
0answers
26 views

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 ...
2
votes
1answer
45 views

If a net is too big, convergence tells us very little?

I'm just learning about nets. It occurs to me that sometimes $x_{\alpha} \to x$ doesn't tell us much at all. Consider the directed set $J:= \{(U,x)\in \mathcal{P}(X) \times X: x \in U \}$, $(U,x) ...
3
votes
2answers
144 views

Is every continuous function measurable?

In non-Hausdorff topology it is standard to define the Borel algebra of a topological space $X$ as the $\sigma$-algebra generated by the open subsets and the compact saturated subsets. Recall that a ...
2
votes
1answer
29 views

Continuity and Leximin

Consider a relation $\geq$ over the set of real-valued vectors. We say that $\geq$ is continuous if for any positive integer $n$, and any $\pi\in\mathbb Z^n,u\in\mathbb R ^n$ we have that the sets ...
1
vote
1answer
35 views

Existence of net of positive real numbers

Does for every totally ordered infinite set $T$ with no greatest element, there exist a net $(a_t )_{t\in T}$ of positive real numbers which is convergent to $0.$ ?
1
vote
0answers
30 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 ...
3
votes
1answer
36 views

Not having c.c.c. implies existence of sequence of open sets with strictly increasing closures?

Suppose a topological space $X$ does not satisfy the countable chain condition (by that I mean that there exists an uncountable family of pairwise disjoint non-empty open subsets of $X$). Does it ...
5
votes
1answer
127 views

Two topologies over the same set which are carried to each other by a lattice isomorphism induce homeomorphic spaces?

This is a question that I've been turning in my head and haven't been able to come up with a way to proceed to either prove it or construct a counter-example: Let $X$ be a set and $\mathfrak{T}_X$ ...
1
vote
0answers
24 views

Does an homeorphism function maps an interval in X to an interval in Y?

Let $(X,\leq_X)$ and $(Y,\leq_Y)$ be two well-ordered sets. Let $f:X\rightarrow Y$ be an homeorphism between $X$ and $Y$ and their order topology (relative to $\leq_X$ and $\leq_Y$). My question is, ...
2
votes
2answers
313 views

Strictly Convex and Strictly Monotonic Preferences

On the assumptions for a well behaved preference, I read recently that a preference must be complete, transitive, continuous, strictly monotonous (if the vectors x≥y and x≠y, then x is strictly ...
2
votes
1answer
117 views

The implications of Completeness and the Continuity axiom for utility representation

Completenes means that every basket of goods in some set previously defined is comparable with the use of a complete preference. Now, with the additional assumption that the preferences are ...
3
votes
3answers
107 views

Topology inducing Order

We know by Munkres that any (total) ordering induces a topology and (luckily) for the real line that coincides with the euclidean topology. In fact, this construction can be carried over to any ...
2
votes
2answers
55 views

Is there a single well-ordered set whose discrete space equals to its order topology?

Let $(X,\leq_X )$ and $(Y,\leq_Y)$ be two well-ordered sets, such that the discrete space of $X$ is the same as the order topology on $(X,\leq_X)$ and the same goes for $Y$. I'd like to prove that ...
0
votes
2answers
79 views

Not well-ordered sets satisfying the least upper bound property

Before proving that every closed interval in $\mathbb{R}$ is compact, James R. Munkres remarks that, in Section 27 entitled "Compact Subspaces of the Real Line" of Topology (2nd edition), Remark: ...
3
votes
1answer
74 views

Uncountable sets and interval topology

Let be an uncountable well-ordered set $(X,\leq)$. We suppose that exists some $x\in X$ such that $y < x$ for uncountably many values of $y$ and define $x_1$ as the minimum $x\in X$ such that $y ...
1
vote
1answer
50 views

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 ...
2
votes
1answer
67 views

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 ...
2
votes
1answer
60 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 ...
3
votes
1answer
104 views

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 ...
1
vote
1answer
40 views

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 ...
1
vote
1answer
33 views

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, ...
0
votes
2answers
82 views

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
1
vote
1answer
208 views

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 ...
5
votes
0answers
108 views

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. ...
1
vote
1answer
101 views

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 ...
0
votes
1answer
73 views

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 $\preceq$ ...
0
votes
1answer
91 views

Definition of initial segment

the definition of initial segment is correct: "let $\preceq$ be a linear ordering of a set $A$, and $B \subsetneqq A$, $B$ is initial segment of $A$ under $\preceq$ if $\forall a \in A, \forall b \in ...
2
votes
1answer
117 views

Completion of a linear order that is a dense subspace of a compact space.

Suppose $D$ is a linearly ordered space which is densely embedded in a compact Hausdorff space $K$. What can we say about the relation between $K$ and $\overline D$, the completion of $D$. Is one a ...
0
votes
0answers
41 views

Interval in Product Space

I am reading a paper now that refers to an "interval" in $\mathbb{R}^{[0,1]}$. But what does interval here refer to? Does this mean there is some ordering of the elements in $\mathbb{R}^{[0,1]}$? I am ...
1
vote
1answer
93 views

Is there a topology with respect to which a poset is connected iff its topologically connected?

A poset $P$ is said to be connected iff for all $x,y \in P$ we can 'zigzag' from $x$ to $y$ in finitely many steps. That is, iff for all $x,y \in P$ there exists a finite sequence $a : n \rightarrow ...
6
votes
2answers
124 views

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 ...
0
votes
1answer
384 views

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.
8
votes
7answers
760 views

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 ...
3
votes
1answer
140 views

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 ...
6
votes
1answer
97 views

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 ...
1
vote
1answer
64 views

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 ...
1
vote
2answers
160 views

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 ...
0
votes
1answer
19 views

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 ...
7
votes
1answer
290 views

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 ...
2
votes
1answer
196 views

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 ...
1
vote
1answer
237 views

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 ...
2
votes
1answer
110 views

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 ...
0
votes
0answers
123 views

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 ...
-2
votes
1answer
126 views

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 ...
1
vote
1answer
114 views

Are homeomorphisms order-isomorphisms?

Is every homeomorphism between topological spaces an order isomorphism (for orders of inclusion $\subseteq$ of sets)?
5
votes
2answers
75 views

Can such a function exist?

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 ...