Questions tagged [order-topology]

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

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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] \...
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70 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 ...
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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 ...
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how to find the limit point?

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 \...
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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 \...
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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 ...
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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 ...
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60 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\...
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Let Y be an ordered set in the order topology. Let $f,g:X→Y$ be continuous. Show that the set $[{x|f(x)≤g(x)}]$ is closed in X.

Let Y be an ordered set in the order topology. Let $f,g:X→Y$ be continuous. Show that the set $[{x|f(x)≤g(x)}]$ is closed in X. My Try : $[{x|f(x)>g(x)}]=∪_{y∈Y}[{x|f(x)>y>g(x)}]$ and $[{x|...
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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 ...
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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 ...
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51 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 $\...
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134 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)...
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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 ...
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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 ...
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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 ...
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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 ...
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111 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 ...
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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 ...
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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 $(...
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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 ...
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247 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 \...
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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 ...
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$X$ 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&...
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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 ...
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1answer
158 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_{\...
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222 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-...
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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 ...
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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 $A=(a,x)$, ...
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Usual and order topology coincides

Show that usual and order topology on $\mathbb R$ coincides. And what does the word "coincides" mean in the statement?
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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....
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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 ...
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361 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....
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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 ...
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109 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 ...
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166 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. ...
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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)) ...
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Claim $(\mathbb{N}, \leq)$ is a discrete space, but is $(-\infty, b)$ a subbasic element?

Let $\mathbb{N}$ denote the set of natural numbers, then a subbasis on $\mathbb{N}$ is $$S = \{(-\infty, b), b \in \mathbb{N}\} \cup \{(a,\infty), a \in \mathbb{N}\}$$ Let $\leq$ be the relation on ...
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Are the order topology on the natural numbers and the discrete toplogy equivalent?

I'm currently trying to get a better understanding of topology and I've read that the order topology on $\mathbb Z$ is equivalent to the discrete topology, since every subset of $\mathbb Z$ is open. ...
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1answer
158 views

Is this proof for Theorem 16.4 Munkres Topology correct?

The followings is the Theorem $16.4$ from Munkres' Topology: In the textbook it uses concept of subbasis to prove the theorem which I can't understand it. I tried to prove that in another way but I ...
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Refrence for order topology

Can anyone give reference for order topology which covers order topology in detail with many examples other than Munkres?
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Order topology vs Subspace topology

$X$: ordered space $Y$: subset of $X$ If $Y$ is not a convex subset of $X$, the order topology of $Y$ and the subspace topology of $Y$ need not be the same. Example: If $X=\mathbb{R}, Y=[0,1) \cup \...
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Topology on the set of ordinal numbers

This is a problem I encountered while reading Topology : An Outline for a First Course by Lewis E. Ward. Suppose $\Omega$ denotes the smallest ordinal number with uncountably many predecessors. Let $\...
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Suborderable space, orderable characterization proof doubt

In Orderability in the presence of local compactness, Valentin Gutev states and proves the following proposition: A suborderable space $X$ is orderable with respect to a linear order $\prec$ on it if ...
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397 views

Open set in order topology

How the set ${x}×[0,1]$ where $x \in [0,1]$ is open in order topology of $\mathbb R^2$ defined by $(x_1,y_1)<(x_2,y_2)$ if EITHER $x_1<x_2$ OR $x_1=x_2$ and $y_1<y_2$. I know that this set ...
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1answer
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How come this is surjective?

Reference: Hocking & Young - Topology p.55 Let $(X,\tau)$ be a compact connected metric space. Let $\leq_X$ be a total ordering on $X$ which induces $\tau$ as the order topology. Let $D$ be a ...
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Why is this intersection a singleton?

Reference: Hocking and Young -Topology p.54 I need to state some definitions and theorems to describe where I get stuck. Definition Let $p,q$ be points of a connected space $S$. We denote $E(...
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Why does there exist a such maximal chain in order topology?

Hocking and Young-Topology p.54 Let $X$ be a topological space equipped with an order topology and $x\in X$. Define $S=\{(a,b): x\in (a,b)\}$ and assume that $S$ is nonempty. How do I show that ...
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Total ordering on $\mathcal P(\Bbb R)$

Is there a total ordering on $\mathcal P(\Bbb R)$, the set of all subsets of $\Bbb R$, such that the set of countable subsets is dense in it? (Given a total ordering $(X,>)$, a set $A\subseteq X$ ...
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Why is a convex subspace the requirement for equivalence beween subspace and order topologies?

I'm currently studying topology, and in one of the lectures we were presented with a theorem that went something like this (rephrasing since I don't have the theorem in front of me): Let $(X, <)$...