Homework Help: Topology Question

1. Nov 14, 2007

emob2p

1. The problem statement, all variables and given/known data
I am asked to show that T=[particular point topology on R^2 ((0,0) being the particular point)] is equal to T'=[topology on R^2 from taking the product of R in the particular point topology (0 being particular point) with itself].

3. The attempt at a solution

I'm pretty sure the book is wrong and this problem can't be solved. Consider the set {(0,0),(1,1)}. This is open in T, but I cannot see how this is open in T' since for the point (1,1) we cannot construct an open set that contains (1,1) but is contained in {(0,0),(1,1)}. The closest we can get is {0,1}x{0,1}={(0,0),(1,0),(0,1),(1,1)}. Am I correct, or am I missing something?

2. Nov 14, 2007

HallsofIvy

For my own benefit, a set is open in the "particular point topology" if and only if it is either empty of contains the "particular point". In particular, a set in R^2 with the particular point topology, (0,0) being the "particular point", T, if and only if it contains (0,0).
A set is open in the "Cartesian product topology" if and only if it is the Cartesian product of two open sets. Here a set is open in T' if and only if it contains a pair (0,y) and the pair (x, 0).

If I have understood this correctly, yes you are right!

3. Nov 14, 2007

emob2p

Not quite, the "Cartesian product topology" is generated from the set of Cartesian products of two open sets. If we took your definition for open sets in R^2, then it would be possible for the union of two open sets to not be open. See the difference?