Steen and Seebach in "Counterexamples in Topology" give "Bernstein's Connected Sets" as item 124.
"Let $\{C_\alpha|\alpha\in [0,\Gamma)\}$ be the collection of all nodegenerate closed connected subsets of the Euclidean plane $\mathbb R^2$ well-ordered by $\Gamma$, the least ordinal equivalent to $c$, the cardinal of the continuum. We define by transfinite induction two nested sequences $\{A_\alpha\}_\alpha < \Gamma$ and $\{B_\alpha\}_\alpha < \Gamma$ such that $A_\alpha \cup B_\alpha = \varnothing$ for all pairs $\alpha, \beta$ ..." etc.
The question is: which Bernstein is this? I'm guessing Felix (as in Cantor-Schroeder-Bernstein etc.) but I'm not completely certain it is. There are few citations for this construct. Hocking and Young mention it, I believe, but I can't access this directly and only see bits of it through the metaphorical keyhole of Google Books.
There's a F. Bernstein citation in the back of S&S: "Zur theorie der trigonometrische Reihe" which I've found originally appeared in Crelle (1907), but I haven't been able to see enough of it to see whether it actually does describe this space.
So, Felix Bernstein here? Or some other (perhaps Sergei)?
That is also the paper by Bernstein cited by Hocking & Young, who give the example on p.110. Steen & Seebach have a note to that item saying that the construction is due to Hocking & Young, who modified an idea of Bernstein. Bernstein’s paper is accessible here [PDF].