All Questions
Tagged with gn.general-topology order-theory
90 questions
27
votes
3
answers
2k
views
When does a Galois connection induce a topology?
Let $(X,\leq)$ and $(Y,\leq)$ by partially ordered sets. Recall that a(n antitone) Galois connection between $X$ and $Y$ is a pair of order-reversing maps
$\Phi: X \rightarrow Y, \ \Psi: Y \...
- 65.4k
24
votes
2
answers
1k
views
Which are the rigid suborders of the real line?
Which are the rigid suborders of the real line?
If A is any set of reals, then it can be viewed as an order structure itself under the induced order (A,<). The question is, when is this structure ...
- 236k
20
votes
2
answers
1k
views
An order type $\tau$ equal to its power $\tau^n, n>2$
(This is a re-post of my old unanswered question from Math.SE)
For purposes of this question, let's concern ourselves only with linear (but not necessarily well-founded) order types.
Recall that:
$...
- 6,819
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 ...
- 41.8k
18
votes
3
answers
1k
views
Is there a natural measurable structure on the $\sigma$-algebra of a measurable space?
Let $(X, \Sigma)$ denote a measurable space. Is there a non-trivial $\sigma$-algebra $\Sigma^1$ of subsets of $\Sigma$ so that $(\Sigma, \Sigma^1)$ is also a measurable space?
Here is one natural ...
- 8,512
18
votes
1
answer
11k
views
Is every continuous function measurable?
This question has already been asked on Math StackExchange here, but was too old to be migrated, and I think will be more appropriate to MathOverflow.
In non-Hausdorff topology it is standard to ...
- 321
13
votes
2
answers
690
views
How many subsets of $\mathbb{R}$ are order isomorphic to $\mathbb{Q}$?
How many subsets of $\mathbb{R}$ are order isomorphic to $\mathbb{Q}$?
How many subsets of the long line $\omega_1\times[0,1)$ are order isomorphic to $\mathbb{Q}$?
I can see that results in both ...
- 233
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 ...
- 307
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 $T_1$-...
- 48.9k
9
votes
1
answer
372
views
Is an open map with open relative diagonal necessarily a local homeomorphism?
Let $f : X \to Y$ be an open (and continuous) map of locales. Suppose the relative diagonal $\Delta_f : X \to X \times_Y X$ is an open embedding of locales. Does it follow that $f : X \to Y$ is a ...
- 15.9k
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 ...
- 48.9k
8
votes
1
answer
657
views
What should the morphisms in the Category of Directed Sets be?
Directed sets are defined to be sets equipped with a preorder that admit (finitary) upper bounds e.g. pairs $(D, \preceq)$ such that $\forall p,q \in D$ there exists $r \in D$ such that $p \preceq r$ ...
- 352
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 ...
- 151
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 ...
- 48.9k
6
votes
1
answer
223
views
Minimal Hausdorff topologies compatible with a bunch of functions
Let $X$ be an infinite set, let ${\cal F}$ be a set of functions $f: X\to X$. We say that a topology $\tau$ is compatible with ${\cal F}$ if every $f\in {\cal F}$ is a continuous function $f:(X, \tau)\...
- 48.9k
6
votes
1
answer
216
views
Can There be Rudin-Keisler Immediate Sucessors?
There are several well-studied orderings on the set $\omega^*$ of ultrafilters on the natural numbers. Three popular ones are $\le_i$ for $i = 1,2,3$. We define $\mathcal U \le_i \mathcal V$ to mean ...
- 1,955
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 ...
- 41.8k
6
votes
0
answers
715
views
What is the structure of a space of $\sigma$-algebras?
Let $X$ be a compact metric space, and consider the Banach space $\Omega = C(X,\mathbb R)$ of continuous, real-valued functions on $X$, equipped with the supremum norm. Let $\delta_x \in \Omega^*$ be ...
- 8,512
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 = \...
- 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 ...
- 476
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-...
- 48.9k
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 ...
- 32.5k
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 ...
- 263
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?
- 48.9k
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_\...
- 48.9k
4
votes
1
answer
218
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 ...
- 48.9k
4
votes
2
answers
225
views
Image of poset with Hausdorff interval topology
Given a poset $(P,\leq)$ the interval topology $\tau_{\text{int}}(P)$ on $P$ is generated by
$$\{P\setminus\downarrow x : x\in P\} \cup \{P\setminus\uparrow x : x\in P\},$$
where $\downarrow x = \{y\...
- 48.9k
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 ...
- 48.9k
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 ...
- 48.9k
4
votes
1
answer
118
views
"Discrete jumps" in the collection of all topologies on a set $X$
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 ...
- 48.9k
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 ...
- 6,541
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\...
- 48.9k
4
votes
0
answers
98
views
Unique representability of bounded distributive lattices
Priestley Duality assigns to every bounded distributive lattice $L$ a compact totally order-disconnected topological space $P(L)$, also called a Priestley space.
A poset $(P,\leq)$ is called (...
- 48.9k
3
votes
2
answers
432
views
When is a filter generated by a (countable) chain?
In any partial order $(P,\leq)$ it is easy to see that every chain generates (i.e., by taking the upwards closure) a filter, and any filter built as a result of the Rasiowa-Sikorski lemma in forcing ...
- 3,115
3
votes
3
answers
1k
views
Countable atomless boolean algebra covered by a larger boolean algebra
Suppose $Q$ is an atomless countable boolean algebra, and $B$ is an arbitrary atomless boolean algebra. $Q$ is unique modulo isomorphisms. There is a subalgebra in $B$ that is isomorphic to $Q$. There ...
- 355
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?
- 48.9k
3
votes
2
answers
252
views
Product of posets with Hausdorff interval topology
Given a poset $(P,\leq)$ the interval topology on $P$ is generated by
$$\{P\setminus\downarrow x : x\in P\} \cup \{P\setminus\uparrow x : x\in P\},$$
where $\downarrow x = \{y\in P: y\leq x\}$ and $\...
- 48.9k
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-...
- 48.9k
3
votes
1
answer
133
views
Path-connected interval topologies
Let $(P,\leq)$ be a poset with more than $1$ point such that the interval topology $\tau_i(P)$ is path-connected. Does this imply that $[0,1]$ order-embeds into $P$?
- 48.9k
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 $$\Big\{P\setminus (\downarrow x): x\in P\Big\} \cup \Big\{P\...
- 48.9k
3
votes
2
answers
320
views
Topological characterisations of properties of posets
Finite connected partially ordered sets are in bijective correspondence to connected finite topological spaces that satisfy T_0, see for example the Wikipedia article Finite topological space. Here ...
- 26.5k
3
votes
1
answer
116
views
Hausdorff interval topology on distributive lattices
Given a poset $(P,\leq)$ the interval topology $\tau_i(P)$ on $P$ is generated by
$$\{P\setminus\downarrow x : x\in P\} \cup \{P\setminus\uparrow x : x\in P\},$$
where $\downarrow x = \{y\in P: y\leq ...
- 48.9k
3
votes
1
answer
318
views
Properties of the interval topology of the lattice of functions
Let $(P,\leq)$ be a poset. The interval topology $\tau_i(P)$ on $P$ is generated by
$$\{P\setminus\downarrow x : x\in P\} \cup \{P\setminus\uparrow x : x\in P\},$$
where $\downarrow x = \{y\in P: y\...
- 48.9k
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....
- 2,585
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,\...
- 48.9k
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 ...
- 48.9k
3
votes
1
answer
107
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 ...
- 48.9k
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 (“...
- 32.5k
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 ...
- 48.9k
3
votes
0
answers
52
views
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 $\kappa$, ...
- 2,830