Is there a "Jordan Big Curve" theorem?

Question

Let the "Big Circle" denote the one-point compactification of the long line, and the "Big Plane" the Cartesian product of the long line with itself. Define a "Jordan Big Curve" to be an injective continuous embedding of the big circle into the big plane. Does an analogue of the Jordan Curve Theorem hold?

Answer 1

I don't think a Jordan Big Curve exists. If you had one, consider its projections to the two "coordinate axes". These projections are compact (because the big circle is), so they lie within a bounded segment of the long line. Enlarge the segment a little to an open interval, and recall that such intervals in the long line are homeomorphic to the real line. So your embedding can be regarded as embedding the big circle in the ordinary plane. The image in the plane of the "compactification point" of the circle has (like any point in the plane) a countable neighborhood base in the plane, but the compactification point in the big circle doesn't. That prevents the embedding from being a homeomorphism. But a continuous injective map between compact Hausdorff spaces (like the big circle and its image in the plane) is always a homeomorphism, so we have a contradiction.