Properties of the product topology
Hello friends. I am doing an independent study course, and it is a bit of the Moore method style. So, right now I am studying product topology and have come up with some conjectures. I have "proof" for all of them but would appreciate (no need for proof if you don't want) if someone could validate whether or not they are true.
1. Let be a class of topological spaces and let be an open base for for each . Then, is an open base for under the product topology.
2. Consequently, if are a finite collection of second countable topological spaces then is separable.
3. If is a collection of topological spaces and is the product space of these spaces, then for any we have that if is a neighborhood of then is a neighborhood of for each
4. The converse is true if is finite.
5. Using this we can show that if is a collection of topological spaces, and is dense in for each then is dense in with the product topology.
6. Consequently, if are separable topological spaces then is separable.
That's it for now.
Any input would be incredibly appreciated. Also, I feel as though I should point out that even though I said this is Moore method like...this is just for my own learning. There is no attempt at foul play here.