Is there such a thing as the *longer* line? This is in reference to the long ray(Alexandroff line) which can be extended in both directions to form the so-called Long Line defined by:
$$ L_{1} = \{  \omega _{1} \times[0,1) \}\cup \{  \omega _{1} \times[-1, 0) \} $$
So  the long plane would be defined as:
$$L_{1} \times L_{1}$$
A similar object is mentioned in the wikipedia article on the bagpipe theorem.
As for the "longer line"(or maybe I should call it the long long line) would be defined as:
$$L_{2} = \{  \lambda _{2} \times[0,1) \}\cup \{  \lambda _{2} \times (-1, 0] \}$$ where  $ \lambda_{2} $ is a limit ordinal whose cardinality is $\beth_ {2}$ (we know such a thing exists as it belongs to a class of transitive sets know as L). So the idea something that inherits the order topology of the real line but has cardinality larger than the continuum. Is there is another name for the longer line as I defined it what is it referred to?
For reference, I found this post by Seewoo Lee describing what he called the "long long line" but it's defined differently and still has the same cardinality of the long line.
Editing Notes: As others have pointed out, the cardinality of
$ \omega _{2}$ is only guaranteed to be larger than the continuum if the GCH holds in the model of set theory we are using. However, using transitive models and the powerset operation we can construct ordinals that have cardinality larger than the continuum in all models of ZF set theory. And so $\lambda _{2}  =\omega _{2} \leftrightarrow GCH \ holds$.
 A: Consider what happens in the point around $\omega_1$, i.e. a "neighborhood" like $[\Gamma_0, \omega_1 + 0.5)$. In a certain sense, the part of this interval from $[\Gamma_0, \omega_1)$ is "denser" than that of $[\omega_1, \omega_1 + 0.5)$. That is, this interval is not uniformly "dense".
What do I mean by "more dense"? In particular, you can have an uncountably long and not-eventually constant sequence accumulating at $\omega_1$ from the left, but in the real numbers, any uncountably long sequence accumulating at a point must be eventually constant, i.e. it actually reaches the point. So there are like "lots more" points "crammed against" $\omega_1$ on the left that the sequence can keep going into to avoid actually having to get there.
That doesn't mean the space isn't "real", of course. It just means it breaks one of the usual motivations for talking about the original long line which is that around every point it still looks like a segment of $\mathbb{R}$. This one, though, doesn't look that way around every point, as I just showed.
