All Questions
30 questions
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
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 ...
- 48.9k
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
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 ...
- 870
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
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 \;\&\; ...
- 639
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
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
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$$
Throughout this discussion I shall refer to ...
- 263
9
votes
3
answers
407
views
Does the lattice of all topologies embed into the lattice of $T_1$-topologies?
Let $\kappa$ be an infinite cardinal, and let $\text{Top}(\kappa)$ be the lattice of all topologies on $\kappa$, ordered by $\subseteq$. Let $\text{Top}^{T_1}(\kappa)$ be the lattice of all $T_1$-...
- 48.9k
3
votes
1
answer
146
views
Maximal elements in the partially ordered set of image spaces
If $(X,\tau)$ is a topological space, let $\text{Im}(X)$ denote the collection of subsets $S$ of $X$ such that there is a continuous function $f:X\to X$ with $\text{im}(f) = S$.
Is there a space $(X,\...
- 48.9k
3
votes
1
answer
143
views
The Wallman and interval topologies on non-principal ultrafilters with the Rudin-Keisler preorder
If $(P,\leq)$ is a pre-odered set (that is, $\leq$ is a reflexive and transitive relation) and $x\in P$, we set $(\uparrow_{\leq} x) = \{p\in P: p\geq x\}$ and $(\downarrow_{\leq} x) = \{p\in P: p\leq ...
- 48.9k
2
votes
1
answer
161
views
Adjoints of the interval topology functor
Given a poset $(P,\leq)$ the interval topology $\tau_i(P)$ on $P$ is generated by
$$\{P\setminus{\downarrow x} : x\in P\} \cup \{P\setminus{\uparrow x} : x\in P\},$$
where $\downarrow x = \{y\in P: y\...
- 48.9k
3
votes
1
answer
116
views
Hausdorff interval topology on distributive lattices
Given a poset $(P,\leq)$ the interval topology $\tau_i(P)$ on $P$ is generated by
$$\{P\setminus\downarrow x : x\in P\} \cup \{P\setminus\uparrow x : x\in P\},$$
where $\downarrow x = \{y\in P: y\leq ...
- 48.9k
6
votes
1
answer
153
views
Topologies with no minimal $T_2$ topologies above them
Let $(X,\tau)$ be a topological space. With $T_2(\tau)$ we denote the collection of $T_2$-topologies on $X$ that contain $\tau$.
Is there an example of a topology $\tau$ such that the partially ...
- 48.9k
4
votes
1
answer
101
views
Spaces that are invariant under some contractions
Let $(X,\tau)$ be a topological space. If $A\subseteq X$ we define the following equivalence relation on $X$: $$\sim_A = \{(x,y) \in X^2: x=y \text{ or }\{x,y\}\subseteq A \}.$$
Let $(X,\tau)$ be an ...
- 48.9k
3
votes
2
answers
133
views
$T_2$-spaces with order-isomorphic topologies
Suppose $X\neq \emptyset$ is a set. Let $\tau_1, \tau_2$ be Hausdorff topologies on $X$ with the property that the partially ordered sets $(\tau_1,\subseteq)$ and $(\tau_2,\subseteq)$ are order-...
- 48.9k
0
votes
1
answer
127
views
Product topology and order convergence topology
Let $(P,\leq)$ be a poset. We define the order convergence topology, denoted by $\tau_o(P)$. By a set filter $\mathcal{F}$ on $P$ we mean a collection of subsets of $P$ such that:
$\emptyset \notin \...
- 48.9k
4
votes
1
answer
368
views
$\mathbb{R}$ and the order-convergence topology
On most partially ordered sets, the order-convergence topology (defined below) is often highly disconnected, often even discrete or [extremally disconnected].1
However, the order-convergence ...
- 48.9k
4
votes
0
answers
152
views
Interval topology of the poset of all coverings
Let $(P,\leq)$ be a poset. The interval topology $\tau_i(P)$ on $P$ is generated by
$$\{P\setminus\downarrow x : x\in P\} \cup \{P\setminus\uparrow x : x\in P\},$$
where $\downarrow x = \{y\in P: y\...
- 48.9k
3
votes
1
answer
163
views
Compactification of order-disconnected spaces
A totally order-disconnected space (TOD) is a tuple $(P, \leq, \tau)$ where $(P, \leq)$ is a poset and $(P,\tau)$ is a topological space such that for $x\not\leq y$ in $P$ there is a clopen down-set ...
- 48.9k
1
vote
1
answer
176
views
Interval topology on complete Boolean algebras
Let $(P,\leq)$ be a poset. The interval topology $\tau_i(P)$ on $P$ is generated by
$$\{P\setminus\downarrow x : x\in P\} \cup \{P\setminus\uparrow x : x\in P\},$$
where $\downarrow x = \{y\in P: y\...
- 48.9k
0
votes
1
answer
183
views
Partially ordered set of compatible topologies
Let $X\neq \emptyset$ be a set and let ${\cal J} \subseteq {\cal P}(X)\setminus\{\emptyset\}$ be a collection of non-empty subsets of $X$. We say that a topology $\tau$ on $X$ is ${\cal J}$-compatible ...
- 48.9k
6
votes
1
answer
223
views
Minimal Hausdorff topologies compatible with a bunch of functions
Let $X$ be an infinite set, let ${\cal F}$ be a set of functions $f: X\to X$. We say that a topology $\tau$ is compatible with ${\cal F}$ if every $f\in {\cal F}$ is a continuous function $f:(X, \tau)\...
- 48.9k
0
votes
1
answer
114
views
Priestley topologizability and connected components
This question is in the spirit of another older question.
We say that a poset $(P,\leq)$ is Priestley-topologizable if there is a topology $\tau$ on $P$ such that $(P,\leq,\tau)$ is a Priestley space....
user70876
1
vote
1
answer
254
views
Interval topology and order convergence topology
Throughout this post, let $(P,\leq)$ be a poset. The interval topology $\tau_i(P)$ on $P$ is generated by
$$\{P\setminus\downarrow x : x\in P\} \cup \{P\setminus\uparrow x : x\in P\},$$
where $\...
user62017
0
votes
2
answers
148
views
Continuous image relation on topological spaces
Let $\kappa$ be a cardinal, and let $\text{Top}(\kappa)$ be the set of topological spaces $(X,\tau)$ such that $X\subseteq \kappa$. We pre-order $\text{Top}(\kappa)$ by
for $X, Y \in \text{...
- 48.9k
4
votes
2
answers
225
views
Image of poset with Hausdorff interval topology
Given a poset $(P,\leq)$ the interval topology $\tau_{\text{int}}(P)$ on $P$ is generated by
$$\{P\setminus\downarrow x : x\in P\} \cup \{P\setminus\uparrow x : x\in P\},$$
where $\downarrow x = \{y\...
- 48.9k
3
votes
2
answers
252
views
Product of posets with Hausdorff interval topology
Given a poset $(P,\leq)$ the interval topology on $P$ is generated by
$$\{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