# 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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### A second countable GO-space can be homeomorphically embedded in a second countable LOTS

A GO-space (generalized ordered space) is a topological space $X$ with topology $\tau$ together with a linear order $<$ such that $X$ is $T_1$ and every point has a local base of $\tau$-open ...
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### Equivalent definitions for GO-spaces (generalized ordered spaces)

GO-spaces (= generalized ordered spaces) are subspaces of LOTS (linearly ordered topological spaces). There are several definitions in use and I am wondering how to show the equivalence between them. ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### Is every ordered field a topological field?

Let $F$ be an ordered field and give $F$ the order-topology. Then is $F$ a topological field (that is, are the operations \begin{split} +&:F\times F\to F\\ -&:F\to F\\ \times&...
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### 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 ...
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### 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?
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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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