- #1
emptyboat
- 28
- 1
Hello, everyone.
Theorem) If each space Xa(a∈A) is a Hausdorff space, then X=∏Xa is a Hausdorff space in both the box and product topologies.
I understand if a box topology, the theorem holds.
but if a product toplogy, I do not understand clearly.
I think if there are distinct points c,d in X, then Uc, Ud (arbitrary open sets in X contain c, d respectively) are equals Xa except for finitely many values of a, so Uc and Ud are not disjoint.
If I have a mistake, please point out it...
Theorem) If each space Xa(a∈A) is a Hausdorff space, then X=∏Xa is a Hausdorff space in both the box and product topologies.
I understand if a box topology, the theorem holds.
but if a product toplogy, I do not understand clearly.
I think if there are distinct points c,d in X, then Uc, Ud (arbitrary open sets in X contain c, d respectively) are equals Xa except for finitely many values of a, so Uc and Ud are not disjoint.
If I have a mistake, please point out it...