Can the definition of "The Long Line" be clarified?

In Steen and Seebach's "Counterexamples in Topology", we see the definition of the Long Line (counterexample 45).

"The long line $L$ is constructed from the ordinal space $[0, \Omega)$ (where $\Omega$ is the least uncountable ordinal) by placing between each ordinal $\alpha$ and its successor $\alpha + 1$ a copy of the unit interval $I = (0,1)$. $L$ is then linearly ordered, and we give it the order topology."

Having given this a bit of thought, I need clarifying the following.

Are the ordinals $0, 1, 2, \ldots, \alpha, \alpha + 1, \ldots$ part of the space, or is $L$ just $\Omega$ instances of $(0,1)$ concatenated? If the latter, then it appears there may be a homeomorphism between $L$ and $[0,\Omega) \times (0,1)$ under the lexicographic ordering. If the former, then it is very much less simple.

So is $L$ like: $0, (0,1), 1, (0,1), 2, (0,1), \ldots, (0,1), \alpha, (0,1), \alpha + 1, (0,1), \ldots, (0,1), \Omega-1, (0,1)$

or is it like:

$(0,1), (0,1), (0,1), \ldots, (0,1), (0,1), (0,1), \ldots, (0,1), (0,1)$

with $\Omega$ instances of $(0,1)$?

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