All Questions
Tagged with gn.general-topology order-theory
84
questions
1
vote
1
answer
289
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 ...
- 167
2
votes
2
answers
269
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 ...
- 25.4k
2
votes
0
answers
88
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 ...
- 830
10
votes
0
answers
249
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 ...
- 317
2
votes
0
answers
59
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 ...
- 28.7k
3
votes
1
answer
196
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 (“...
- 28.7k
5
votes
2
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475
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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 = \...
- 28.7k
5
votes
1
answer
159
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 ...
- 476
2
votes
0
answers
50
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 $f_{i}$ ...
- 5,275
2
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0
answers
36
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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 ...
- 589
0
votes
1
answer
136
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 \;\&\; ...
- 629
3
votes
1
answer
143
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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....
- 2,495
1
vote
1
answer
129
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 ...
- 261
1
vote
1
answer
140
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 $$R^{-1}(V) = \{u\in U: \exists v\in V:(u,v)\...
- 44.5k
0
votes
1
answer
108
views
Ordering preserved by an inverse frame homomorphism
Recall that a frame homomorphism $h:L\to M$ is called ($L$ and $M$ are frames):
Dense if, for any $x ∈ L$, $h(x) = 0$ implies $x = 0$.
Codense if, for any $x ∈ L$, $h(x) = 1$ implies $x = 1$.
...
5
votes
2
answers
364
views
Non-homeomorphic connected $T_2$-spaces with isomorphic topology poset
What are examples of non-homeomorphic connected $T_2$-spaces $(X_i,\tau_i)$ for $i=1,2$ such that the posets $(\tau_1, \subseteq)$ and $(\tau_2,\subseteq)$ are order-isomorphic?
- 44.5k
4
votes
1
answer
226
views
Measurable total order
Under what conditions on a metric space $X$, equipped with the Borel $\sigma$-algebra, does there exist a measurable total ordering of the elements of $X$?
By "measurable total ordering" we ...
- 5,949
1
vote
2
answers
176
views
Reference request: lower sets of a preorder form a lattice
Consider a set $S$ with a preorder $\preceq$ (a preorder is a reflexive and transitive relation). A lower set $A$ of $S$ is defined as a subset of $S$ such that for all $x \in S$ and $y \in A$, if $...
- 610
2
votes
1
answer
180
views
Basis or subbasis for Scott topology
Let $X$ be a partially ordered set. A subset $S\subseteq X$ is called Scott-open if and only if it is:
Upward-closed: $x\in S$ and $x\le y$ implies $y\in S$;
Inaccessible by directed suprema: if $D\...
- 2,099
2
votes
1
answer
115
views
Is the Scott topology generated by the ideals as the closed sets?
Let $X$ be a directed-complete partial order, or even a complete lattice. A subset $S\subseteq X$ is called Scott-closed if and only if it is:
Downward-closed: $y\in S$ and $x\le y$ implies $x\in S$;
...
- 2,099
2
votes
1
answer
229
views
Fixed point property and interval topology
Given a poset $(P,\leq)$ the interval topology $\tau_i(P)$ on $P$ is generated by
$$\{P\setminus\downarrow x : x\in P\} \cup \{P\setminus\uparrow x : x\in P\},$$
where $\downarrow x = \{y\in P: y\leq ...
- 44.5k
4
votes
1
answer
208
views
Embedding ordinals with the order topology into connected $T_2$-spaces
Is there a limit ordinal $\kappa_0$ with $\kappa_0 \lt 2^{\aleph_0}$ and such that for every limit ordinal $\lambda$ with $\kappa_0\leq \lambda\lt 2^{\aleph_0}$ there is a connected $T_2$-space $X_\...
- 44.5k
1
vote
2
answers
210
views
Can the Boolean Algebra of regular open sets be isomorphic to ${\cal P}(\omega)/(\text{fin})$?
Let $(X,\tau)$ be a topological space. $A\subseteq X$ is said to be regular open if $A = \text{int}(\text{cl}(A))$ and let $\text{RO}(X,\tau)$ denote the collection of regular open sets of $X$. A ...
- 44.5k
-5
votes
1
answer
304
views
Borromean rings, Condorcet's paradox and Quantum chromodynamics [closed]
In https://plus.google.com/108432079989441783124/posts/LHewqvcj5Xo T. Abderrahman explains what Borromean rings are. As I noticed in a comment, the underlying order structure is the same as in ...
- 6,954
2
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1
answer
130
views
Topologically Ordered Families of Disjoint Cantor Sets in $I$?
Suppose that we have an uncountable collection $C_\alpha$ of disjoint Cantor Sets contained in the closed unit interval $I$. Suppose we have ordered the indices $\alpha \in [0,1]$ as well. Then is ...
- 409
4
votes
1
answer
252
views
Order convergence vs topological convergence in partially ordered sets
Short version of the question. If $(P,\leq)$ is a partially ordered set (poset), a topology denoted by $\tau_o(P)$ can be defined (see below). There is also another notion of convergence, called order-...
- 44.5k
5
votes
0
answers
160
views
(When) is the Dedekind-MacNeille completion of a po-set Hausdorff?
Let $X$ be a p.o. Consider the topology on $X$ generated by
$$U_{x}^{-}:=X\setminus (x\uparrow),\quad U_{x}^{+}:=X\setminus (x\downarrow), \quad x\in X$$
Throughout this discussion I shall refer to ...
- 263
6
votes
0
answers
113
views
Closedness of the partial order in complete Hausdorff semitopological semilattices
First some definitions.
A semilattice is a commutative semigroup consisting of idempotents (i.e., elements such that $xx=x$). A typical example of a semilattice is the unit interval endowed with the ...
- 40.2k
8
votes
2
answers
204
views
Spaces without maximal homogeneous subspaces
A homogeneous space $(X,\tau)$ is a topological space such that for all $x,y\in X$ there is a homeomorphism $\varphi:X\to X$ such that $\varphi(x)=y$. As a previous question implies, the union of an ...
- 44.5k
9
votes
3
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404
views
Does the lattice of all topologies embed into the lattice of $T_1$-topologies?
Let $\kappa$ be an infinite cardinal, and let $\text{Top}(\kappa)$ be the lattice of all topologies on $\kappa$, ordered by $\subseteq$. Let $\text{Top}^{T_1}(\kappa)$ be the lattice of all $T_1$-...
- 44.5k
3
votes
1
answer
143
views
Maximal elements in the partially ordered set of image spaces
If $(X,\tau)$ is a topological space, let $\text{Im}(X)$ denote the collection of subsets $S$ of $X$ such that there is a continuous function $f:X\to X$ with $\text{im}(f) = S$.
Is there a space $(X,\...
- 44.5k
3
votes
1
answer
142
views
The Wallman and interval topologies on non-principal ultrafilters with the Rudin-Keisler preorder
If $(P,\leq)$ is a pre-odered set (that is, $\leq$ is a reflexive and transitive relation) and $x\in P$, we set $(\uparrow_{\leq} x) = \{p\in P: p\geq x\}$ and $(\downarrow_{\leq} x) = \{p\in P: p\leq ...
- 44.5k
2
votes
1
answer
158
views
Adjoints of the interval topology functor
Given a poset $(P,\leq)$ the interval topology $\tau_i(P)$ on $P$ is generated by
$$\{P\setminus{\downarrow x} : x\in P\} \cup \{P\setminus{\uparrow x} : x\in P\},$$
where $\downarrow x = \{y\in P: y\...
- 44.5k
3
votes
1
answer
113
views
Hausdorff interval topology on distributive lattices
Given a poset $(P,\leq)$ the interval topology $\tau_i(P)$ on $P$ is generated by
$$\{P\setminus\downarrow x : x\in P\} \cup \{P\setminus\uparrow x : x\in P\},$$
where $\downarrow x = \{y\in P: y\leq ...
- 44.5k
6
votes
1
answer
151
views
Topologies with no minimal $T_2$ topologies above them
Let $(X,\tau)$ be a topological space. With $T_2(\tau)$ we denote the collection of $T_2$-topologies on $X$ that contain $\tau$.
Is there an example of a topology $\tau$ such that the partially ...
- 44.5k
4
votes
1
answer
99
views
Spaces that are invariant under some contractions
Let $(X,\tau)$ be a topological space. If $A\subseteq X$ we define the following equivalence relation on $X$: $$\sim_A = \{(x,y) \in X^2: x=y \text{ or }\{x,y\}\subseteq A \}.$$
Let $(X,\tau)$ be an ...
- 44.5k
3
votes
2
answers
126
views
$T_2$-spaces with order-isomorphic topologies
Suppose $X\neq \emptyset$ is a set. Let $\tau_1, \tau_2$ be Hausdorff topologies on $X$ with the property that the partially ordered sets $(\tau_1,\subseteq)$ and $(\tau_2,\subseteq)$ are order-...
- 44.5k
2
votes
2
answers
223
views
What's "serialization" really called, and is there any theory surrounding it?
Define an operator $\mathop{\vec{\bigcup}}$ as follows:
Definition. Whenever $A$ is an $I$-indexed family of sets, where $I$ is a totally-ordered set, we have $$\mathop{\vec{\bigcup}}_{i \in I} A_i ...
- 3,683
0
votes
1
answer
117
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 \...
- 44.5k
4
votes
1
answer
359
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 ...
- 44.5k
4
votes
0
answers
149
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\...
- 44.5k
19
votes
1
answer
448
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 ...
- 40.2k
7
votes
2
answers
890
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 ...
- 141
3
votes
1
answer
159
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 ...
- 44.5k
1
vote
1
answer
170
views
Interval topology on complete Boolean algebras
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\...
- 44.5k
2
votes
1
answer
244
views
What lattices are isomorphic to $R^{N}$ for some $N$, equipped with the product order?
What lattices are isomorphic to $\mathbb{R}^{N}$ for some $N\in \mathbb{N}$, equipped with the canonical order?
Remark:
When I say $\mathbb{R}^N$, I don’t mean it to be a vector space. Instead, I ...
- 63
3
votes
0
answers
130
views
Duality for continuous lattices based on [0, 1]
A continuous lattice may be defined as a complete lattice in which arbitrary meets distribute over directed joins. A continuous lattice is naturally regarded as an algebraic structure where the ...
- 133
0
votes
1
answer
159
views
Partially ordered set of compatible topologies
Let $X\neq \emptyset$ be a set and let ${\cal J} \subseteq {\cal P}(X)\setminus\{\emptyset\}$ be a collection of non-empty subsets of $X$. We say that a topology $\tau$ on $X$ is ${\cal J}$-compatible ...
- 44.5k
4
votes
1
answer
212
views
Minimal zero-dimensional Hausdorff spaces
A topological space $(X,\tau)$ is said to be zero-dimensional Hausdorff (zdH) if for $x\neq y\in X$ there is $C\subseteq X$ clopen (closed and open) such that $x\in C$, but $y\notin C$.
We say a zdH ...
- 44.5k
3
votes
1
answer
106
views
Topology with no direct lower neighbor
Given any poset $(P,\leq)$ and $x, y\in P$ we set $[x,y] = \{p\in P: x\leq p \leq y\}$. For any set $X$, let $\text{Top}(X)$ denote the set of topologies on $X$. The set $\text{Top}(X)$ is a complete ...
- 44.5k