All Questions
Tagged with order-theory general-topology
319
questions
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53
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Extending the $M_3,N_5$ theorem from distributive lattices to frames
It is known that a lattice $L$ is distributive if and only if it does not contain the diamond $M_3$ or the pentagon $N_5$ as sublattices.
A complete lattice is one in which every subset has an infimum ...
- 53
2
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1
answer
92
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Convexity structures and partial orders
Can any convexity structure be defined by a partial order $\preceq$ in the sense of the order topology: a given set $A$ is convex if for any $a,b \in A$ and any other element $c$ for which $a\preceq c ...
- 438
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Reconciling Continuity of Binary Relations with Continuity of Functions/Correspondences
I asked this question in the Economics StackExchange as well, but figured it may be better-suited here.
There are various ways to express the concept of continuity of a binary relation, but one I've ...
- 91
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78
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Complete iff Compact in Well-Ordered space
Let $T=(S, \leq, \tau)$ a well-ordered set equipped with order topology (defined here).
Definition 1: $T$ is called complete iff every non-empty subset of $T$ has a greatest lower bound (inferior) and ...
- 188
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Equivalent definitions for GO-spaces (generalized ordered spaces)
GO-spaces (= generalized ordered spaces) are subspaces of LOTS (linearly ordered topological spaces). There are several definitions in use and I am wondering how to show the equivalence between them.
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Every second countable LOTS is embeddable in $\mathbb R$
I'd like to prove the following (Engelking, exercise 6.3.2(c)).
Theorem: Every second countable LOTS is embeddable in $\mathbb R$.
Here, LOTS = linearly ordered topological space. "Embeddable ...
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5
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164
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Is subset relation preserved under limit for Hausdorff metric?
Let $X$ be a metric space. I consider elements in $Y=2^X\setminus \emptyset$ and use the Hausdorff metric for $Y$.
Suppose that $A_n \subseteq B_n$ for $A_n,B_n \in Y$ and $A_n \rightarrow A$ and $B_n ...
- 111
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3
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197
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Are linearly ordered topological spaces well-based?
A linearly-ordered topological space or LOTS is one whose topology admits a basis generated by open intervals of a total ordering of its points.
A well-based space is one which admits a local basis of ...
- 153
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41
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Necessity of denseness and completeness for a surjective monotone being continuous
It turns out that the familiar result that surjective monotones $\mathbb R\to\mathbb R$ are continuous extends to general LOTS (linearly ordered topological spaces):
Theorem. If $X$, $Y$ are LOTS ...
- 4,119
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56
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The subspace topology of $Y_u$($Y$ with upper topology) is strictly coarser than the one induced from $X$?
Let $(X,\leqslant )$ be a poset, we define the upper topology has the principle upper sets, that is upper sets of the form $\left \{ \uparrow x:x\in P\right \} $, as the subbase. We can define ...
- 31
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60
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How do you prove that all countable, densely ordered sets without endpoints are isomorphic to the rationals?
I've looked online for a proof of this and have found several references to Canter's isomorphism theorem and the "back-and-forth method." However, I haven't been able to find any explicit ...
- 25
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3
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73
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LOTS and radial properties for generalized Fort/Fortissimo spaces
This question explores the linearly orderable and radial properties of Fort and Fortissimo spaces and their generalizations.
A Fort space is a set $X$ with a distinguished point, call it $\infty$, and ...
- 4,500
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45
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What is exactly is a continuous curve in a Lorentzian manifold?
Often, a continuous function $f: \mathbb{R} \rightarrow M$ from the real numbers $\mathbb{R}$ to a metric space $M$ is called continuous at $r \in \mathbb{R}$ if and only if, for any $\epsilon>0$, ...
- 137
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42
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If the numbers in one set are all larger than those in another set, are there numbers in between?
Let $S, R \subseteq \mathbb{R}$ be sets of real numbers such that for every $s \in S$ and $r \in R$, $s<r$. How can one prove that there exists some real number $t$ satisfying $s <t <r$ for ...
- 137
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72
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Topology as an order (not order topology)
Given a topological space $(X,T)$, the topology $T$ is also a partial order with the inclusion relation $(T,\subseteq)$.
Given a continuous function $f:A\to B$ between two spaces $(A,T_1)$ and $(B, ...
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