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Does there exist a multi-valued "monotone" and "compact" map from a Boolean algebra to the "free" part of $\mathcal{P}(\kappa)$?

This is a follow-up to my previous question, which has a negative answer. Here is the most general version that I'm interested: Does there exist a Boolean algebra $A$, an infinite cardinal $\kappa$, ...
David Gao's user avatar
  • 2,830
3 votes
1 answer
132 views

Is the interval topology on ${\cal P}(\omega)/(\text{fin})$ connected?

If $(P,\leq)$ is a poset and $x\in X$, we let $\downarrow x = \{p\in P: p \leq x\}$, and $\uparrow x$ is defined dually. The collection $$\Big\{P\setminus (\downarrow x): x\in P\Big\} \cup \Big\{P\...
Dominic van der Zypen's user avatar
0 votes
0 answers
62 views

Order-convergence and interval topology on ${\cal P}(\omega)/(\text{fin})$

On any poset $(P, \leq)$ we can consider two different topologies that arise directly from the ordering relation. 1) Order convergence topolog $\tau_o(P)$ : By a set filter $\mathcal{F}$ on $P$ we ...
Dominic van der Zypen's user avatar
5 votes
1 answer
95 views

Preimage of a sublocale by a morphism of locales: description by nucleus?

For completeness of MathOverflow, and to avoid any possible misunderstanding, let me recall the following terminology and facts, which should be standard (experts skip the following 2–3 paragraphs ...
Gro-Tsen's user avatar
  • 32.5k
0 votes
0 answers
42 views

Topologizing quasi orders with regards to products

This morning I was asked by a colleague for the "right" way to construct a topology on a quasi-order (aka preorder, a reflexive and transitive relation) such that the topology on a product ...
Steven Clontz's user avatar
3 votes
1 answer
119 views

Non-isomorphic $T_0$-spaces with order-isomorphic topologies

Are there non-isomorphic $T_0$-spaces $(X_i, \tau_i)$ for $i = 1,2$ such that $\tau_1 \cong \tau_2$ when considered as partially ordered sets?
Dominic van der Zypen's user avatar
1 vote
1 answer
344 views

Is there anyway to formulate the Alexandrov topology algebraically?

One knows that the Alexandrov topology on a preordered set is the finest topology that induces the same [specialization] preorder on the set. Given this, one finds a one-to-one correspondence between ...
Bastam Tajik's user avatar
3 votes
2 answers
320 views

Topological characterisations of properties of posets

Finite connected partially ordered sets are in bijective correspondence to connected finite topological spaces that satisfy T_0, see for example the Wikipedia article Finite topological space. Here ...
Mare's user avatar
  • 26.5k
2 votes
0 answers
92 views

Can this order relation, defined in terms of all topological spaces, be defined in terms of the reals alone?

Let $K$ be the operator monoid under composition of Kuratowski's $14$ set operators generated by topological closure $k$ and complement $c.$ Kuratowski's 1922 paper gives the poset diagram of the ...
mathematrucker's user avatar
10 votes
0 answers
265 views

Let $X$ be a finite set of $n$ ($>1$) elements and $\tau$ be a topology on $X$ having exactly $m$ elements. Can we give any description of $m$?

Let $X$ be a finite set of $n$ ($>1$) elements and $\tau$ be a topology on $X$ having exactly $m$ elements. Can we give any description of $m$ as it relates to $n$? Obviously $2\le m\le 2^n$ and ...
SoG's user avatar
  • 307
2 votes
0 answers
105 views

What is known about sublocales defined by regular nuclei?

(For basic terminology, which is supposed to be standard anyway, see this other question, which inspired this one.) I am interested in nuclei $j\colon L\to L$ on a frame $L$ which are regular elements ...
Gro-Tsen's user avatar
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3 votes
1 answer
228 views

Computing the Heyting operation on the frame of nuclei

(The following definitions are meant to be standard and are reproduced for completeness of the question.) A frame is a partially ordered set in which every finite subset has a greatest lower bound (“...
Gro-Tsen's user avatar
  • 32.5k
5 votes
2 answers
496 views

Do germs of open sets around a point form a frame?

Let $X$ be a topological space and $x \in X$ a point. Let $\Omega$ be the set of open sets (viꝫ. the topology) of $X$, and $\Omega_x$ the set of germs around $x$ of open sets, that is, $\Omega_x = \...
Gro-Tsen's user avatar
  • 32.5k
5 votes
1 answer
162 views

Scott topology: Suprema of sequences are topological limits

I read that, with the Scott topology, suprema of sequences are topological limits (See page 1 of this article). Let $(X, \le)$ be a DCPO, and $D$ be a directed subset of $X$. I can easily see that the ...
Bob's user avatar
  • 476
2 votes
0 answers
52 views

Can we decompose an increasing net of functions into two increasing nets with prescribed supports?

Let $K$ be a compact Hausdorff space and let $U,V\subset K$ be open. Let $\left(f_{i}\right)_{i\in I}$ be an increasing net of continuous non-negative functions such that $f_{i}\le 1$ and $f_{i}$ ...
erz's user avatar
  • 5,529
3 votes
0 answers
43 views

Continuous analogue for Szpilrajn Theorem: complete preorder extends a continuous preorder

A corollary of Szpilrahn Theorem states: Any preorder on nonempty $X$ has a complete and transitive extension. I am thinking about the "Szpilrahn Theorem" for continuous preorder on ...
dodo's user avatar
  • 599
0 votes
1 answer
159 views

Partial orders on downward closed sets [closed]

Let $P = (V, \sqsubseteq)$ be a partial order and $\mathfrak{D}(P)$ denote the class of downward-closed subsets of the partial order $P$ (i.e, the class of $A \subseteq V$ such that $y\in A \;\&\; ...
user65526's user avatar
  • 639
3 votes
1 answer
162 views

A closed subset of a Dedekind-complete order has subspace topology equal to order topology

Here's a fairly easy fact from point-set topology that I'm having trouble finding a reference for. Say $X$ is a total order satisfying the least-upper bound property, and $S$ is a closed subset of it....
Harry Altman's user avatar
  • 2,585
1 vote
1 answer
137 views

Density and compactness of Boolean embeddings

Let A and B be Boolean algebras and $h:A\rightarrow B$ a Boolean embedding. If every element of $B$ can be expressed both as a join of meets and as a meet of joins of elements in $h(A)$, then the ...
IJM98's user avatar
  • 281
1 vote
1 answer
143 views

Is the Rudin-Keisler ordering a continuous relation?

If $X, Y$ are topological, and $R\subseteq X\times Y$ we say that $R$ is continuous (from $X$ to $Y$) if for every $V\subseteq Y$ with $V$ open, we have $$R^{-1}(V) = \{u\in U: \exists v\in V:(u,v)\...
Dominic van der Zypen's user avatar
0 votes
1 answer
129 views

Ordering preserved by an inverse frame homomorphism

Recall that a frame homomorphism $h:L\to M$ is called ($L$ and $M$ are frames): Dense if, for any $x ∈ L$, $h(x) = 0$ implies $x = 0$. Codense if, for any $x ∈ L$, $h(x) = 1$ implies $x = 1$. ...
Biller Alberto's user avatar
4 votes
2 answers
385 views

Non-homeomorphic connected $T_2$-spaces with isomorphic topology poset

What are examples of non-homeomorphic connected $T_2$-spaces $(X_i,\tau_i)$ for $i=1,2$ such that the posets $(\tau_1, \subseteq)$ and $(\tau_2,\subseteq)$ are order-isomorphic?
Dominic van der Zypen's user avatar
4 votes
1 answer
236 views

Measurable total order

Under what conditions on a metric space $X$, equipped with the Borel $\sigma$-algebra, does there exist a measurable total ordering of the elements of $X$? By "measurable total ordering" we ...
Aryeh Kontorovich's user avatar
1 vote
2 answers
195 views

Reference request: lower sets of a preorder form a lattice

Consider a set $S$ with a preorder $\preceq$ (a preorder is a reflexive and transitive relation). A lower set $A$ of $S$ is defined as a subset of $S$ such that for all $x \in S$ and $y \in A$, if $...
Artemy's user avatar
  • 695
2 votes
1 answer
227 views

Basis or subbasis for Scott topology

Let $X$ be a partially ordered set. A subset $S\subseteq X$ is called Scott-open if and only if it is: Upward-closed: $x\in S$ and $x\le y$ implies $y\in S$; Inaccessible by directed suprema: if $D\...
geodude's user avatar
  • 2,129
2 votes
1 answer
143 views

Is the Scott topology generated by the ideals as the closed sets?

Let $X$ be a directed-complete partial order, or even a complete lattice. A subset $S\subseteq X$ is called Scott-closed if and only if it is: Downward-closed: $y\in S$ and $x\le y$ implies $x\in S$; ...
geodude's user avatar
  • 2,129
2 votes
1 answer
233 views

Fixed point property and interval topology

Given a poset $(P,\leq)$ the interval topology $\tau_i(P)$ on $P$ is generated by $$\{P\setminus\downarrow x : x\in P\} \cup \{P\setminus\uparrow x : x\in P\},$$ where $\downarrow x = \{y\in P: y\leq ...
Dominic van der Zypen's user avatar
4 votes
1 answer
221 views

Embedding ordinals with the order topology into connected $T_2$-spaces

Is there a limit ordinal $\kappa_0$ with $\kappa_0 \lt 2^{\aleph_0}$ and such that for every limit ordinal $\lambda$ with $\kappa_0\leq \lambda\lt 2^{\aleph_0}$ there is a connected $T_2$-space $X_\...
Dominic van der Zypen's user avatar
1 vote
2 answers
235 views

Can the Boolean Algebra of regular open sets be isomorphic to ${\cal P}(\omega)/(\text{fin})$?

Let $(X,\tau)$ be a topological space. $A\subseteq X$ is said to be regular open if $A = \text{int}(\text{cl}(A))$ and let $\text{RO}(X,\tau)$ denote the collection of regular open sets of $X$. A ...
Dominic van der Zypen's user avatar
-5 votes
1 answer
313 views

Borromean rings, Condorcet's paradox and Quantum chromodynamics [closed]

In https://plus.google.com/108432079989441783124/posts/LHewqvcj5Xo T. Abderrahman explains what Borromean rings are. As I noticed in a comment, the underlying order structure is the same as in ...
Sylvain JULIEN's user avatar
2 votes
1 answer
133 views

Topologically Ordered Families of Disjoint Cantor Sets in $I$?

Suppose that we have an uncountable collection $C_\alpha$ of disjoint Cantor Sets contained in the closed unit interval $I$. Suppose we have ordered the indices $\alpha \in [0,1]$ as well. Then is ...
John Samples's user avatar
5 votes
1 answer
284 views

Order convergence vs topological convergence in partially ordered sets

Short version of the question. If $(P,\leq)$ is a partially ordered set (poset), a topology denoted by $\tau_o(P)$ can be defined (see below). There is also another notion of convergence, called order-...
Dominic van der Zypen's user avatar
5 votes
0 answers
171 views

(When) is the Dedekind-MacNeille completion of a po-set Hausdorff?

Let $X$ be a p.o. Consider the topology on $X$ generated by $$U_{x}^{-}:=X\setminus (x\uparrow),\quad U_{x}^{+}:=X\setminus (x\downarrow), \quad x\in X$$ Throughout this discussion I shall refer to ...
Thomas's user avatar
  • 263
6 votes
0 answers
117 views

Closedness of the partial order in complete Hausdorff semitopological semilattices

First some definitions. A semilattice is a commutative semigroup consisting of idempotents (i.e., elements such that $xx=x$). A typical example of a semilattice is the unit interval endowed with the ...
Taras Banakh's user avatar
  • 41.8k
8 votes
2 answers
205 views

Spaces without maximal homogeneous subspaces

A homogeneous space $(X,\tau)$ is a topological space such that for all $x,y\in X$ there is a homeomorphism $\varphi:X\to X$ such that $\varphi(x)=y$. As a previous question implies, the union of an ...
Dominic van der Zypen's user avatar
9 votes
3 answers
407 views

Does the lattice of all topologies embed into the lattice of $T_1$-topologies?

Let $\kappa$ be an infinite cardinal, and let $\text{Top}(\kappa)$ be the lattice of all topologies on $\kappa$, ordered by $\subseteq$. Let $\text{Top}^{T_1}(\kappa)$ be the lattice of all $T_1$-...
Dominic van der Zypen's user avatar
3 votes
1 answer
146 views

Maximal elements in the partially ordered set of image spaces

If $(X,\tau)$ is a topological space, let $\text{Im}(X)$ denote the collection of subsets $S$ of $X$ such that there is a continuous function $f:X\to X$ with $\text{im}(f) = S$. Is there a space $(X,\...
Dominic van der Zypen's user avatar
3 votes
1 answer
143 views

The Wallman and interval topologies on non-principal ultrafilters with the Rudin-Keisler preorder

If $(P,\leq)$ is a pre-odered set (that is, $\leq$ is a reflexive and transitive relation) and $x\in P$, we set $(\uparrow_{\leq} x) = \{p\in P: p\geq x\}$ and $(\downarrow_{\leq} x) = \{p\in P: p\leq ...
Dominic van der Zypen's user avatar
2 votes
1 answer
161 views

Adjoints of the interval topology functor

Given a poset $(P,\leq)$ the interval topology $\tau_i(P)$ on $P$ is generated by $$\{P\setminus{\downarrow x} : x\in P\} \cup \{P\setminus{\uparrow x} : x\in P\},$$ where $\downarrow x = \{y\in P: y\...
Dominic van der Zypen's user avatar
3 votes
1 answer
116 views

Hausdorff interval topology on distributive lattices

Given a poset $(P,\leq)$ the interval topology $\tau_i(P)$ on $P$ is generated by $$\{P\setminus\downarrow x : x\in P\} \cup \{P\setminus\uparrow x : x\in P\},$$ where $\downarrow x = \{y\in P: y\leq ...
Dominic van der Zypen's user avatar
6 votes
1 answer
153 views

Topologies with no minimal $T_2$ topologies above them

Let $(X,\tau)$ be a topological space. With $T_2(\tau)$ we denote the collection of $T_2$-topologies on $X$ that contain $\tau$. Is there an example of a topology $\tau$ such that the partially ...
Dominic van der Zypen's user avatar
4 votes
1 answer
101 views

Spaces that are invariant under some contractions

Let $(X,\tau)$ be a topological space. If $A\subseteq X$ we define the following equivalence relation on $X$: $$\sim_A = \{(x,y) \in X^2: x=y \text{ or }\{x,y\}\subseteq A \}.$$ Let $(X,\tau)$ be an ...
Dominic van der Zypen's user avatar
3 votes
2 answers
133 views

$T_2$-spaces with order-isomorphic topologies

Suppose $X\neq \emptyset$ is a set. Let $\tau_1, \tau_2$ be Hausdorff topologies on $X$ with the property that the partially ordered sets $(\tau_1,\subseteq)$ and $(\tau_2,\subseteq)$ are order-...
Dominic van der Zypen's user avatar
2 votes
2 answers
249 views

What's "serialization" really called, and is there any theory surrounding it?

Define an operator $\mathop{\vec{\bigcup}}$ as follows: Definition. Whenever $A$ is an $I$-indexed family of sets, where $I$ is a totally-ordered set, we have $$\mathop{\vec{\bigcup}}_{i \in I} A_i ...
goblin GONE's user avatar
  • 3,793
0 votes
1 answer
127 views

Product topology and order convergence topology

Let $(P,\leq)$ be a poset. We define the order convergence topology, denoted by $\tau_o(P)$. By a set filter $\mathcal{F}$ on $P$ we mean a collection of subsets of $P$ such that: $\emptyset \notin \...
Dominic van der Zypen's user avatar
4 votes
1 answer
368 views

$\mathbb{R}$ and the order-convergence topology

On most partially ordered sets, the order-convergence topology (defined below) is often highly disconnected, often even discrete or [extremally disconnected].1 However, the order-convergence ...
Dominic van der Zypen's user avatar
4 votes
0 answers
152 views

Interval topology of the poset of all coverings

Let $(P,\leq)$ be a poset. The interval topology $\tau_i(P)$ on $P$ is generated by $$\{P\setminus\downarrow x : x\in P\} \cup \{P\setminus\uparrow x : x\in P\},$$ where $\downarrow x = \{y\in P: y\...
Dominic van der Zypen's user avatar
19 votes
1 answer
465 views

Large Borel antichains in the Cantor cube?

Let $2^\omega$ be the Cantor cube $\{0,1\}^\omega$, endowed with the standard compact metrizable topology and the standard product measure, called the Haar measure. The Cantor cube is considered as a ...
Taras Banakh's user avatar
  • 41.8k
8 votes
2 answers
944 views

When does Scott topology generated by specialization order induced by a sober space (X,$\tau$) equal the initial topology $\tau$?

Let X be a $T_{0}$ space. The specialization order ≤ on X is that if x is contained in cl{y}, then we call "x≤y". Obviously (X,≤) is a partially ordered set. A sober space is a topological space such ...
Zhenchao Lyu's user avatar
3 votes
1 answer
163 views

Compactification of order-disconnected spaces

A totally order-disconnected space (TOD) is a tuple $(P, \leq, \tau)$ where $(P, \leq)$ is a poset and $(P,\tau)$ is a topological space such that for $x\not\leq y$ in $P$ there is a clopen down-set ...
Dominic van der Zypen's user avatar