Re: Complement of zero dimensional space
- From: "G. A. Edgar" <edgar@xxxxxxxxxxxxxxxxxxxxxxxxxxx>
- Date: Fri, 26 Jun 2009 07:49:28 -0400
In article <20090626021218.A68276@xxxxxxxxxxxxxxx>, William Elliot
<marsh@xxxxxxxxxxxxxxxx> wrote:
Conjecture. If S is a zero dimensional subspace of R^2,
then R^2 - S is a dense, path connected subspace.
An example is zero dimensional S = QxP \/ PxQ
and path connected R^2 - S = Q^2 x P^2. (P = R\Q)
Trivial examples are any countable S (regular countable
spaces are zero dimensional) and path connected R^2 - S.
A counter example to the converse is the set of all points
of rational slope lines, including the y-axis, through (0,0).
Counter examples are always welcome,
because they preempt laboring over a proof.
----
A topological space has topological dimension n if and only if it can
be written as the union of n+1 sets of dimension zero.
R^2 has topological dimension 2.
So: If S is a zero-dimensional subspace of R^2, then R^2 - S has
dimension at least 1. Furthermore, for any nonempty open set U in R^2,
also U - S has dimension at least 1. This proves S is dense in R^2.
Let x,y be distinct points in R^2. Since R^2 has dimension 2, any set
that separates x from y has dimension at least 1. So: R^2 - S is
connected.
This did not do path connected, though.
--
G. A. Edgar http://www.math.ohio-state.edu/~edgar/
.
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