Questions tagged [order-topology]
Ordered sets have a natural topology generated by the open intervals. This tag is meant for questions about this topology.
93
questions
1
vote
0
answers
70
views
Exercise 3, Section 31 of Munkres’ Topology
Show that every order topology is regular.
My attempt:
Approach(1): Let $X$ be an ordered set equipped with $\mathcal{T}_o$ order topology. Let $\{ x\}$ be a singleton set in $X$. Then $(x,+\infty)\...
- 1,359
1
vote
0
answers
28
views
closed set and the supremum in order topology
Suppose $\alpha$ is an ordinal endowed with the order topology, i.e. its basic open sets are generalized open intervals. Given $C \subseteq \alpha$, I want to show (1) implies (2) :
(1) For all ...
- 143
2
votes
1
answer
39
views
Find an element which is not in the topology.
The order topology is defined on $\{0,1\} \times \mathbb{N}$. Need to find a non-singleton set which is not an element of order topology.
I Choose $A=\{(0,10), (1,1)\}$. How to prove this? Is my ...
- 662
0
votes
1
answer
83
views
Prove: If $A$ and $B$ are closed subsets of $[0,\Omega]$ then at least $A$ or $B$ is bounded
As usual, I am self studying topology and my knowledge of ordinals is meagre. Have
done some research on it.
Theorem 5.1 Any countable subset of $[0,\Omega)$ is bounded above.
(This exercise requires ...
user960654
1
vote
1
answer
52
views
Doubt related to example 4 of section 14 chapter 2 of Munkres Topology book
The example goes as follows
The set X = {1,2} × $\mathbb{Z^+}$ in the dictionary order is another example of an ordered set with a smallest element. Denoting $1×n$ by $a_n$ and $2×n$ by $b_n$
We can ...
- 407
3
votes
1
answer
157
views
Is every ordered field a topological field?
Let $F$ be an ordered field and give $F$ an order topology. Then is $F$ a topological field (that is, are the operations
\begin{equation}
\begin{split}
+&:F\times F\to F\\
-&:F\to F\\
\times&...
- 1,215
0
votes
1
answer
55
views
Limit points and order topology
Let $X = [0,1) \cup \{2\} \subset \mathbb{R}$. Consider two topologies on X. The subspace topology on X induced by the standard topology on $\mathbb{R}$, and the order topology on X induced by the ...
- 21
0
votes
1
answer
49
views
What is the topology defined on $[0, \omega_1)$ it and how do we define its basis?
The set $[0, \omega_1)$ is a countably compact topological space. How do we prove it to be a topological space. How does its basis elements look like?
- 67
3
votes
3
answers
75
views
If f: $\mathbb{R}_{l} \rightarrow S_{\Omega}$ is continuous, then f is not injective.
If f: $\mathbb{R}_{l} \rightarrow S_{\Omega}$ is continuous, then f is not injective.
I've been trying to solve this problem for a few days, but I haven't been able to see how can I do it. First, $\...
- 55
0
votes
2
answers
72
views
Supremum in a dictionary ordered set.
Take double arrow space $[0,1]\times \{1,2\}$
Where topology is dictionary topology defined by linear order as $(a,b)<(c,d)$ iff $a<c$; or ;$a=c \;\&\; b<d$
open sets are in the form $$((...
- 730
0
votes
1
answer
42
views
Separation axioms of topology
Let $X=Z_+×Z_2, Y=Z_2×Z_+$ with the dictionary order topologies, are $X$ and $Y$,$T_1$,$T_2$ and $T_3$?
$Z_+=\{1,2,3,...\}$
$Z_2 =\{0,1\}$
The open set on the dictionary ordering will be
$I=\{x×y∈\...
1
vote
0
answers
27
views
Conclusion about corollary V.6.1 in Schaeffer & Wolff (Topological Vector Spaces 2nd edition)
Currently, I am stuck with following the final conclusion in the proof of corollary V.6.1 in Schaeffer & Wolff's Topological Vector Spaces, 2nd edition, pp. 230-231. Probably it is trivial and I ...
- 167
2
votes
1
answer
319
views
Questions about completely normal spaces.
I'm trying to solve the next problem: A topological space $(X,\tau)$ is called completely normal if, and only if, every subspace is normal. Prove that the following conditions are equivalent:
a) $X$ ...
- 91
0
votes
4
answers
190
views
Munkres order topology difference between definition of simple order using $<$ instead of $\leq$
Munkres in his Topology 2ed section 14 p. 84 defines a simply ordered set as in this math.stack question here
definition of simply ordered set by Munkres
I recalled that I have come across the ...
- 158
0
votes
2
answers
432
views
What is definition of order topology?
I am doing self study of topology, the book I am following has used term order topology in problems and few theorems. What is order topology? Kindly give explanation with some examples that will be ...
user706475
1
vote
3
answers
254
views
How is the closed ordinal space compact hausdorff?
So the original question in the paper was whether every compact hausdorff space automatically metrizable? I found out that closed ordinal space with order topology is a counter example. Since the ...
- 298
0
votes
1
answer
200
views
Open sets in the order topology
I'm having trouble understanding the order topology. If we consider $R_{\ge 0} \times R_{\ge 0}$ given the order topology coming from the lexicographic order, my understanding is that the sets $[2,3] \...
- 77
0
votes
1
answer
294
views
Linear continuum of $I\times I $ under subspace topology of $\mathbb{R}^2$ with dictionary topology on it, where $I=[a,b]$.
Let $I\times I$ be subspace of space $\mathbb{R}^2$ with dictionary order, where $I=[a,b]$. What can you say about the linear continuum of $I\times I$ with the subspace topology.
{ I've proved that ...
- 63
-1
votes
1
answer
92
views
If Y is a subset of X, is always true that then Y inherits (always) a total order from X?
If Y is a subset of X, then Y inherits a total order from X. The set Y therefore has an order topology, the induced order topology
What does it mean "inherits" ? Is this a a Kripke model M ...
- 5
0
votes
0
answers
30
views
What are the limit points of $X \times \mathbb{N}$?
Let $X=\{0,1,2 \}$ with natural order topology and give $\mathbb{N}$ its natural order topology. Consider $X \times \mathbb{N}$ with the dictionary order topology.
What are the limit points of $X \...
- 9,500
0
votes
1
answer
155
views
Help me to visualise limit points of $S$. [duplicate]
Consider the ordered square $I^2,$ then the set $[0,1]\times [0,1]$ with the dictionary order. Let the general element of $I^2$ be denoted by $x\times y,$ where $x,y\in[0,1].$ The closure of the ...
- 602
0
votes
2
answers
103
views
Order Topology on a Preorder
While looking at the definition of the order topology defined on a total order (https://en.wikipedia.org/wiki/Order_topology), I realized I needed a generalization to preorders. So ultimately the ...
- 171
0
votes
1
answer
73
views
How is this definition of closedness compatible with the order topology?
Let $\kappa$ be a limit ordinal.
Taken from the definition of a closed unbounded set,
we say a subset $C\subseteq\kappa$ is closed in $\kappa$ if and only if
$\sup(C\cap\alpha)=\alpha<\kappa\...
- 87
1
vote
2
answers
172
views
If $f\colon X\to Y$ is continuous and $X$ is compact Hausdorff and connected, $Y$ is given an order topology. Is $f$ necessarily onto?
I want to ask you for following statement
$f$ is continuous from $X$ to $Y$, where $X$ is compact Hausdorff, connected and $Y$ is ordered set in order topology, then $f$ is onto?
My attempt
For ...
- 879
8
votes
2
answers
229
views
Can the long line be embedded in the ordered plane?
It is a well known result that the long line (namely, the topological space $S_\omega \times [0, 1)$ in the order topology, where $S_\omega$ is the minimal uncountable well-ordered set) cannot be ...
- 1,635
0
votes
1
answer
105
views
$\omega_1$ is a limit point of the subset $[0,\omega_1)$
In the Wiki of order topology, I encounter the following statement.
$\omega_1$ is a limit point of the subset $[0,\omega_1)$ even though no sequence of elements in $[0,\omega_1)$ has the element $\...
- 14k
2
votes
2
answers
510
views
Right continuous functions when considered with lower limit topology
Exercise 17.7 (a) from James Munker's Topology says:
Suppose that $f : \mathbb R \rightarrow \mathbb R$ is "continuous from the right", i.e.
$$
\forall_{a \in \mathbb R} \lim_{x \rightarrow a^+} f(x)...
- 1,700
0
votes
1
answer
83
views
Closed subsets of regular cardinals
Let $\kappa$ be a regular cardinal, equipped with order topology, and $C\subseteq\kappa$ a closed unbounded subset.
If $\alpha<\kappa$ is another regular cardinal and a limit point of $C$, is the ...
- 101
4
votes
1
answer
422
views
Munkres Example 16.3
The example says:
Let $I=[0,1]$. The dictionary order on $ I \times I$ is just the restriction to $I\times I$ of the dictionary order on the plane $\mathbb{R} \times \mathbb{R}$. However, the ...
- 558
1
vote
1
answer
170
views
The lub of the usual and co-countable topology on $\mathbb R$
This exercise comes from Steven A. Gaal's Point Set Topology(1964,Academic Press), Page 37, exercise 4. It is
Let $\mathscr T$ be the least upper bound of the usual topology and of the topology ...
- 338
6
votes
2
answers
158
views
Open sets on ordered topology in $\mathbb{R\times R}$
I am having some difficulty grasping the concept of an ordered topology in $\mathbb R \times\mathbb R $. The definition I was given is that this is the topology where $(a,b) < (c,d)$ if $a<c$ or ...
- 153
0
votes
2
answers
200
views
Separating Closed Sets in a Well-Ordered Set in the Order Topology
Theorem 32.4 from Munkres' Topology book:
Every well-ordered set $X$ is normal in the order topology
This is something a continuation of this post, where, evidently, I forgot what the definition ...
- 7,062
3
votes
2
answers
362
views
Prob. 7 (a), Sec. 24, in Munkres' TOPOLOGY, 2nd ed: Any order preserving surjective map between ordered sets is a homeomorphism
Here is Prob. 7 (a), Sec. 24, in the book Topology by James R. Munkres, 2nd edition:
Let $X$ and $Y$ be ordered sets in the order topology. Show that if $f \colon X \to Y$ is order preserving and ...
- 23.7k
0
votes
1
answer
73
views
Is the interval $((a,b), (a,d))$ is open in $\mathbb{R}\times \mathbb{R}$ which is equipped with the dictionary order topology
Is the interval $((a,b), (a,d))$ is open in $\mathbb{R}\times
\mathbb{R}$ which is equipped with the dictionary order topology.
For me, since $(a,b)$ and $(a,c)$ are points in $\mathbb{R^2}$ with $(...
- 7,121
1
vote
1
answer
585
views
questions about subspace topology, order topology and convex subset
I am learning Munkres' s topology, and have some questions about subspace topology, order topology and convex subset.
I have read 3 similar questions here [ref]:One question on subspace topology and ...
- 151
0
votes
1
answer
572
views
Show that R^2 with the dictionary order topology is homeomorphic to Rd * R.
I am currently working topology and I can not prove this exercise.
Let $Rd$ denote the set $R$ with the discrete topology. Show that $R^2$ with the dictionary order topology is homeomorphic to $Rd \...
2
votes
1
answer
375
views
The closure of $(a;b)$ in an order topology [duplicate]
Consider a linearly ordered set $(X; \prec)$ with its order topology.
Show that closure of $(a;b)$ is a subset of $[a;b]$. Under what conditions does equality hold?
Give an example of a strict ...
2
votes
1
answer
695
views
$X$ and $Y$ are ordered set, $f:X\rightarrow Y$ is order preserving and surjective, then $f$ is a homeomorphism.
Let $X$ and $Y$ are ordered set in the order topology. Show that if $f:X\rightarrow Y$ is order preserving and surjective, then $f$ is a homeomorphism.
My attempt:
Simply $f$ is bijective (take $x&...
- 281
0
votes
2
answers
193
views
linear ordered topological spaces are $T_4$- guided proof.
I would like to prove the following exercise, for which my topology book give a hint.
"Let $X$ be a linear ordered topological space, than $X$ is $T_4$".
A space is $T_4$ if for all $A,B$ closed ...
- 319
2
votes
1
answer
259
views
Show the $\omega_1$-line (long line) is not homeomorphic and not order isomorphic to $[0,1)$
This is based on Ex. 6.4.6 in Stillwell's "Real Numbers."
Using previous exercises, it was established that one can construct for any countable ordinal $\gamma$ disjoint half-open intervals $[a_{\...
user12802
1
vote
1
answer
537
views
Existence of continuous order-preserving function
Let $X$ be a countable simple ordered set. I am trying to prove that there exists on X a real, order-preserving function, continuous in the order topology.
I construct by induction an order-...
- 469
4
votes
2
answers
317
views
A totally ordered ring with its order topology is not a topological ring.
There is a theorem that, if $R$ is a totally ordered ring which is also a division ring, then $R$ is a topological division ring with respect to the order topology on $R$. I am certain that, in ...
- 48.4k
0
votes
1
answer
142
views
Linearly ordered X is regular
Prove that every linearly ordered space X is regular.
Can anyone please help me with this proof?
I started with letting $x$ belong to $x$ and take a nbhd $U=(a,b)$ of $x$ and then taking $A=(a,x)$, ...
- 93
1
vote
1
answer
476
views
Is the Function a Homeomorphism?
This an example from a book I am reading. The author claims that there are two ways to see that the bijection $f : (-1,1) \rightarrow \mathbb{R}$ defined by $f(x) = \frac{x}{1-x^2}$ is a homeomorphism....
- 7,062
3
votes
1
answer
182
views
Complete ordered space
In their paper A Method for Constructing Ordered Continua , Hart and van Mill give the following definition of ordered continuum:
An ordered continuum is a compact, connected linearly ordered ...
- 2,846
4
votes
1
answer
976
views
Is $[0,1]^2$ with the dictionary order topology separable? First countable?
I have question about order square $I^2_0$ which is $[0,1]\times[0,1]$ with dictionary order
this is what I know about it: compact Hausdorff, hence normal and regular as well. and also it is Lindelof....
- 810
7
votes
0
answers
273
views
Homeomorphism between evenly spaced integer topology and the rationals
The evenly spaced integer topology is countable, metrizable, and has no isolated points, and hence is homeomorphic to the rationals with the order topology. But what is an explicit construction for ...
- 6,086
0
votes
1
answer
183
views
Continuous surjective function from $[0,1] \rightarrow$ lexicographical order
This is a follow up question to the question I asked here:
The range of a continuous function on the order topology is convex
Let $(X,\mathcal{T})$ denote the unit square with the lexicographical ...
- 825
2
votes
1
answer
291
views
The range of a continuous function on the order topology is convex
Let $(W, \leq)$ be a linear order, and let $f : [0, 1] \rightarrow W$ be a continuous function (where [0, 1] has the usual topology and W has its order topology). Show that the range of f is convex.
...
- 825
2
votes
1
answer
840
views
Is the unit square with dictionary ordering second countable?
I'm conflicted:
If we consider the set $\{x\} \times (0,1)$, for $ x \in [0,1]$, these are open in the unit square, uncountable and disjoint, but what about open intervals of the form ((a,b), (c,d)) ...
- 825