What's application of Michael line? Michael line could see here.
Let $M=\mathbb{R}$ and $\tau_M=\{U\cup A: U$ open in $\mathbb{R}, A\subset \mathbb{R} \setminus \mathbb{Q}\}$. Then $(M,\tau_M)$ is a topological space called the Michael Line.
I need note that the machael line has other forms. For example, we pick another dense subset $S$ of $\mathbb R$, not other $\mathbb R\setminus \mathbb Q$. By the same method, we can obtain another Michael Line.

My question is this: What's  application of Michael line?

Thanks for your any help.
 A: This particular version of the Michael line was discovered (or invented..) to give a negative answer to the question: is the product of a completely metrizable separable space and a paracompact space again paracompact? 
Already it was known that a product of paracompact spaces need not be paracompact, as seen by the square of the Sorgenfrey line. But maybe some theorem is possible if we strengthen the paracompactness of one of the factors? It was already known too that the product of a paracompact space and a compact (Hausdorff) space is again paracompact, so people looked for other nice classes of spaces that would preserve paracompactness in products. Separable complete metric spaces seem quite nice, so that would have been a nice theorem. But it turns out that $M \times \mathbb{P}$ is not even normal, killing all hopes for such a theorem. Note that $\mathbb{P}$, the irrationals, is indeed a completely metrizable space (being a $G_\delta$ in $\mathbb{R}$).
Later the generalization you mentioned was found, and if we base a Michael line $M'$ on a Bernstein set $B \subset \mathbb{R}$, then $B \times M'$ is likewise non-normal (essentially the same argument as for the irrationals) and then we have the product of a separable metric space ($B$) with a Lindelöf regular space (which is strongly paracompact) that is non-normal. So there we strengthen the paracompact space and weaken the completeness, with a similar result. In fact, if you are willing to assume CH (in fact weaker axioms suffice), we can find a Michael line construction $M''$ whose product with the irrationals $\mathbb{P}$ is non-normal, and it is (AFAIK) unknown whether a ZFC example of such a pair of spaces (completely metrizable separable, and regular Lindelöf) exists.
So in short, spaces like this illustrate the boundaries of what we can and cannot hope to prove, in this case mostly in the realm of paracompactness/normality of product spaces.
A lot of this is also explained here, with full proofs.
