# "$X$ is an ordered set in the order topology." What is the meaning of this sentence? (James R. Munkres "Topology 2nd Edition")

I am reading "Topology 2nd Edition" by James R. Munkres.

There is the following sentence on p.90 in this book:

Now let $$X$$ be an ordered set in the order topology, and let $$Y$$ be a subset of $$X$$.

I am very poor at English.
What is the meaning of the following sentence?:

$$X$$ is an ordered set in the order topology.

Does the above sentence mean the following?:

$$X$$ is an ordered set and the topology on $$X$$ is the order topology.

• Yes, that is what it means. Nov 1, 2020 at 10:19
• @JoséCarlosSantos Thank you very much for your answer. Nov 1, 2020 at 10:37

Yes, it means exactly that.

Munkres considers an ordered set $$X$$ and puts the order topology on it.

If you take a subset $$Y \subseteq X$$, there are two possible topologies on $$Y$$:

(1) $$Y$$ is itself an ordered set, so it obtains an order topology as well.

(2) $$Y$$ is a subset of $$X$$, so it obtains a subspace topology.

A natural question is whether these topologies are the same. This turns out to be false, as Munkres demonstrates.

• MathQED, Thank you very much for your answer. Nov 1, 2020 at 10:38
• @tchappyha You are very welcome! Nov 1, 2020 at 10:41