- #1
Mogarrr
- 120
- 6
Homework Statement
This is from Baby Rudin Exercise 2.20- Are closures and interiors of connected sets always connected? (Look at subsets of [itex]\mathbb{R}^2 [/itex]).
Homework Equations
The interior is the set of all interior points for a set E that is a subset of a metric space X.
A subset Y of a metric space X is disconnected if and only if there are open subsets U, V of X satisfying:
1) [itex]U \cap Y \neq \emptyset [/itex], [itex] V \cap Y \neq \emptyset[/itex]
2) [itex] Y \subset U \cap V[/itex]
3) [itex]U \cap V \cap Y = \emptyset [/itex]
The Attempt at a Solution
It's best if I draw a picture, but I was basically thinking of a dumbbell in [itex]\mathbb{R}^2 [/itex], that is two closed circles with a line segment connecting them.
I can prove that the interior of this shape is disconnected, however I am having some trouble showing that the original set, the dumbbell is connected.
This is homework for a class where the teacher takes points off (or fails to give points, you know you're perspective of the half empty/ half full glass) for not having justification.
So how to justify this set, the dumbbell, is connected. I've thought of using proof by contradiction, but I'm not sure where the contradiction will come, and I'm running low on time. Any help would be much appreciated.