Re: Complement of zero dimensional space


On Fri, 26 Jun 2009, G. A. Edgar wrote:
<marsh@xxxxxxxxxxxxxxxx> wrote:

Conjecture. If S is a zero dimensional subspace of R^2,
then R^2 - S is a dense, path connected subspace.

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.

A = [0,1] /\ Q and B = [1,2] /\ Q are two zero dimensional spaces.

Thus C = A \/ B = [1,2] /\ Q is one dimensional?

R^2 has topological dimension 2.

By what definition of dimension?

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 R^2 - S is dense in R^2.

Correction has been made as per your other post.

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.

If T is at lest two dimensional and connected and Z is
zero dimensional, is T - Z dense connected subset of T?


.

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