- #1
PsychonautQQ
- 784
- 10
Let S be the minimal uncountable set. That is, for every m in S, there are countable many n s.t. n<m.
Let L = { S x [0,1) } \ ##(a_0,0)## where ##a_0## is the smallest element of S (S is well ordered so this element exists). Order L be dictionary order, and then give L the order topology.
Show that L is locally euclidean, Hausdorff and path connected.
_____________
I am first trying to understand why L is Hausdorff. If ##a_0## is the smallest element of S, then S/{a_0} will have smallest element a_1.
I am trying to find disjoint neighborhoods of points (a_1,0) and (a_2,0) which I believe to be the two smallest elements of L. Does each element of S have a discrete neighborhood in the order topology?
After I understand why it is Hausdorff then I will try to understand why it is locally euclidean and path connected.
Thanks PF!
Let L = { S x [0,1) } \ ##(a_0,0)## where ##a_0## is the smallest element of S (S is well ordered so this element exists). Order L be dictionary order, and then give L the order topology.
Show that L is locally euclidean, Hausdorff and path connected.
_____________
I am first trying to understand why L is Hausdorff. If ##a_0## is the smallest element of S, then S/{a_0} will have smallest element a_1.
I am trying to find disjoint neighborhoods of points (a_1,0) and (a_2,0) which I believe to be the two smallest elements of L. Does each element of S have a discrete neighborhood in the order topology?
After I understand why it is Hausdorff then I will try to understand why it is locally euclidean and path connected.
Thanks PF!