Questions tagged [order-topology]

Ordered sets have a natural topology generated by the open intervals. This tag is meant for questions about this topology.

Filter by
Sorted by
Tagged with
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)\...
user avatar
  • 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 ...
user avatar
  • 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 ...
user avatar
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 ...
user avatar
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 ...
user avatar
  • 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&...
user avatar
  • 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 ...
user avatar
  • 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?
user avatar
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, $\...
user avatar
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 $$((...
user avatar
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∈\...
user avatar
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 ...
user avatar
  • 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) $X$ ...
user avatar
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 ...
user avatar
  • 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 ...
user avatar
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 ...
user avatar
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] \...
user avatar
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 ...
user avatar
  • 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 ...
user avatar
  • 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 \...
user avatar
  • 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 ...
user avatar
  • 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 ...
user avatar
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\...
user avatar
  • 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 ...
user avatar
  • 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 ...
user avatar
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 $\...
user avatar
  • 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)...
user avatar
  • 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 ...
user avatar
  • 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 ...
user avatar
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 ...
user avatar
  • 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 ...
user avatar
  • 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 ...
user avatar
  • 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 ...
user avatar
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 $(...
user avatar
  • 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 ...
user avatar
  • 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 \...
user avatar
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 ...
user avatar
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&...
user avatar
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 ...
user avatar
  • 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_{\...
user avatar
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-...
user avatar
  • 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 ...
user avatar
  • 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 $A=(a,x)$, ...
user avatar
  • 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....
user avatar
  • 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 ...
user avatar
  • 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....
user avatar
  • 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 ...
user avatar
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 ...
user avatar
  • 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. ...
user avatar
  • 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)) ...
user avatar
  • 825