Why does the author assumes convexity of a subspace while proving the connectivity of a linear continuum?

Question

In the book of Topology by Munkres, at page 153, it is given that

If $L$ is a linear continuum in the order topology, then $L$ is connected and so are intervals and rays in $L$.

And in the proof,

We prove that if $Y$ is a convex subspace of $L$, then $Y$ is connected.

And indeed the author uses the convexity of $Y$ in the proof.However, in the theorem there is no mention of convexity, so I do not understand why does the author assume the convexity of a subspace of a linear continuum while proving the connectivity of that linear continuum ?

Answer 1

Intervals and rays are order-convex, clearly. The whole space is trivially so.

In a connected ordered space, the opposite is also true: all order convex subsets of $X$ are either $X$, an (open or closed) ray, or an (open, closed, half-open) interval.

Answer 2

You say 'in the theorem there is no mention of convexity'.

But a statement about continuum, intervalls and rays in it is more or less exactly that: a Statement about convex objects.

And he is not assuming convexity, but using their common property as tool for the proof.

It is obvious that $L$ is convex, and it's discussed here: Linear continuum is convex