Disconnecting R^n

[William Elliot]

Let E be a finite dimensional Euclidean space. If C is a closed subset
of E that disconnects E, then is it always true that some component of
C also disconnects E?

[David Cullen]

Is there any set that disconnects R^n that has the property that none
of its components disconnect R^n?

Ie is the assumption that such a set be closed known to be essential?

[William Elliot]

I doubt it.

Let q:N -> Q be an enumeration of [0,1] /\ Q,

K = \/{ {a}x(R - {q^-1(a)} | a in [0,1] /\ Q }.

Is R^2 - K disconnected?  Is K connected?

This and similar constructions all seem to fail as
an example to show the necessity of a closed set.

[David Cullen]

But R^2 - K constructed this way is connected.  You can see this
because if you have a disconnection into two open sets U,V then every
set of the form {i}xR with i irrational is contained in one or the
other of U,V.  Now either all such sets are in U in which case we are
done, or one of these sets {j}xR is in V, in which case let s in R be
the supremum over all points x such that (x,0) is in U with x < j.  s
must be rational or else (s,0) is in the boundary of U and the boundary
of V, contrary to U,V being a disconnection.  Still, even if s is
rational, (s, q^-1(s)) is in the boundary of U and the boundary of V.

[William Elliot]

What's K?  That set K previous determined in another thread
to be connected but not path connected.  Please include sufficient
context in your replies so I can know what you're talking about.

Note, some usenet readers are less sophisticated than others.

[David Cullen]

I mean K as defined in the context of the present thread, namely: K =