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

Question

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.

Answer 1

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.