All Questions
Tagged with lattice-orders general-topology
93
questions
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There are more sublocales than subspaces.
In this article: https://arxiv.org/pdf/2406.12486v1 it is written:
Given a topological space $X$, there are typically more
sublocales in its frame of opens $\Omega(X)$ than subspaces in X.
Can you ...
- 35
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52
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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
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29
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If $(y_\lambda)_{\lambda \in \Lambda}$ is a subnet of $(x_\alpha)_{\alpha \in A}$ and if $A$ is a complete lattice, is $\Lambda$ a complete lattice?
Let $X$ be a set, $(x_\alpha)_{\alpha \in A}$ be a net in $X$ and $(y_\lambda)_{\lambda \in \Lambda}$ be a subnet of $(x_\alpha)_{\alpha \in A}$.
If $A$ is a complete lattice (i.e. not only a directed ...
1
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30
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The functor $Sob \to Loc $ is faithful
I want to show that the functor $O: Sob \to Loc$ from the sober topological spaces to the locales is faithful. So take $X,Y$ two sober topological spaces. I want to show that the map
$$O_{X,Y}: Hom(X,...
- 35
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35
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$X\setminus \overline{\{ x\}}$ is meet irreducible in the lattice $O(X)$.
Let $X$ be a topological space and $x\in X$ I want to show that $X\setminus \ \overline{\{ x\}}$ is meet-irreducible in the lattice $O(X)$ of open sets of $X$. This means that we want to show that $X\...
- 35
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1
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43
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The open sets form a complete lattice [duplicate]
Let $X$ be a topological space and denote $O(X)$ the set of open sets of $X$. Then I read that
"$O(X)$ is a complete lattice since the union of any family of open sets is again
open". I don'...
- 11.1k
2
votes
1
answer
45
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Stone-Cech compactification via lattice ideals of $Coz(X)$
While studying P. T. Johnstone's book Stone Spaces I have come across the following Proposition (Section: 3.3)
Where $max C(X)$ is the set of all maximal ideals of $C(X)$ (ring of real-valued ...
- 1,068
3
votes
1
answer
88
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Is there a connection between the topology of $\mathbb{R}/\mathbb{Q}$ and the logic fundamental to Smooth Infinitesimal Analysis?
As I read through the discussion related to this question (Visualizing quotient groups: $\mathbb{R/Q}$) posted 11 years ago I am reminded of a line on page 20 in a book by J. L. Bell, A Primer of ...
2
votes
1
answer
93
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"No infinite discrete subspace" vs "No infinite pairwise disjoint family of opens"
What is the relationship between the two properties (for a topological space $X$)
$A$: "$X$ has no infinite family of pairwise disjoint open subsets"
$B$: "$X$ has no infinite discrete ...
- 38.7k
5
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70
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Lattices/Topology and the Stone Duality
For some context I have some partial understanding of lattices and an intermediate understanding of topology.
I at some point in the past week started thinking about a funny way to view a topology on ...
- 61
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19
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Disjoint product of lattices / the lattice of disjoint pairs of open sets is not a lattice, but is nearly so
Consider a lattice $L=(|L|,\wedge,\vee,\top,\bot)$ and define its disjoint product $|L\otimes L|$ to be a set as follows:
$$
|L\otimes L| = \{(l,l')\in L\times L \mid l\wedge l'=\bot\} .
$$
We see ...
- 538
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votes
1
answer
67
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Is there a situation where any intersection of upward-closed subset is not upward-closed(can be infinite)?
In any quasi-ordered set X, we say that a subset A is upward-closed iff whenever $x\in A \ $and$\ x \leqslant y$, then $y\in A$.
Consider the collection $\mathcal{O}$ of all upward closed subsets of $...
- 31
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75
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Why are nets defined with directed sets (which only requires upper bound to finite subset)
The definition of net in topology is defined via a directed set, that is a set $A$ with a preorder such that every finite subset of $A$ has an upper bound.
If my understanding is correct, an ordered ...
- 1,714
2
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66
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Applications of a theorem of M. H. Stone to general topology
Exercise 2.I of J. Kelley's book General Topology introduces a theorem by M.H. Stone concerning maximal ideals in distributive lattices. The statement is the following :
Let $A$ and $B$ be disjoint ...
- 35.3k
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55
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A special topology compatible with partial orders
Question. Are there contradictions with the following topology $\tau$? If not, is it an established topology known with a conventional name? Thanks in advance.
Suppose $(Y, <)$ is a non-empty ...
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