Connected Spaces

1. Dec 4, 2009

latentcorpse

I've been asked to find disjoint, non-empty, disconnected subspaces $A,B \subset \mathbb{R}$ such that $A \cup B$ is connected.

My problem is in that because the A and B are open and disjoint, when i take the union i keep getting one point omitted which prevents the union from being connected.

i was wondering about $A=\mathbb{Z}$ and $B=\mathbb{R} \backslash \mathbb{Z}$. These are disjoint, non-empty and open subsets of the real line and when u take their union you get $\mathbb{R}$ which is connected. i'm not sure about the disconnectedness of A and B though...

2. Dec 4, 2009

Dick

Z is NOT open. And R is not the disjoint union of two open sets (disconnected or not). The first statement you made of the problem says nothing about A and B being open.

3. Dec 4, 2009

latentcorpse

sry. that was a big mistake. i think i've got it now though, thanks!

4. Dec 5, 2009

HallsofIvy

Staff Emeritus
Did the problem really say "disconnected"? What does that mean? If it just means "disjoint", you don't need to say it. My first thought was to interpret it as "separated" but then this problem is impossible.