A different kind of connectedness
Usually we prove that if a topological space 
is arcwise connected, it is connected by contradiction. We assume 
form a disjoint non-empty open cover of , pick elements from each, and use the path between these elements to construct our contradiction. The same idea holds if instead of having a path between each pair of elements, we have a path between each pair of nonempty open sets– let’s call this setwise connectedness (I made this up; if there’s a standard name, please let me know). Can you find an example of a space that is setwise connected, but not arcwise connected? When does setwise connectedness imply pathwise connectedness?
Possibly relevant posts:
- Lattice Problem (2/16/2005)
- Exactness of differential forms (7/28/2007)
- Complex Analysis pt. I (11/3/2003)
Here’s an example of a space that is setwise connected but not arcwise connected. In the x-y plane, take the union of the line joining (0, -1) and (0, 1) with the graph of sin(1/x) for 0 < x < 1. In the topology induced by the plane, every open set containing part of the y-axis also contains part of the sine curve and hence the space is setwise connected. However, no arc connects a point in the sine curve to a point on the y axis.
Comment by John — 10/12/2008 @ 12:37 pm
Clearly, the dreaded example of the closure of the graph of $\sin(1/x)$ is “set-wise connected”.
Continuing with your thoughts - I always wondered whether this example of connected but not path-connected set is somehow generic or is a part of some classification of such scenarios.
Comment by Torus — 10/12/2008 @ 6:14 pm
Torus, I guess your point is: set-wise connectedness is exactly connectedness! Silly me. I was also aiming for something intermediate on the scale between connected and path-connected. It seems like a big gap.
Comment by Alex — 10/12/2008 @ 9:45 pm
I don’t agree with that - take the closure of the graph of sin(1/fractional value of x) then sets with different integer value will not be set-wise connected.
But this is more of the same kind of example.
Comment by Torus — 10/14/2008 @ 9:23 am