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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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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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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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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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 ...
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 ...