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.

Conjectures:

1. Let $\displaystyle \left\{X_j\right\}_{j\in\mathcal{J}}$ be a class of topological spaces and let $\displaystyle \mathfrak{B}_\ell$ be an open base for $\displaystyle X_\ell$ for each $\displaystyle \ell\in\mathcal{J}$. Then, $\displaystyle \mathfrak{B} = \prod_{j\in\mathcal{J}}\mathfrak{B}_j$ is an open base for $\displaystyle X = \prod_{j\in\mathcal{J}}X_j$ under the product topology.

2. Consequently, if $\displaystyle X_1,\cdots,X_n$ are a finite collection of second countable topological spaces then $\displaystyle X_1\times\cdots\times X_n$ is separable.

3. If $\displaystyle \left\{X_j\right\}_{j\in\mathcal{J}}$ is a collection of topological spaces and $\displaystyle X = \prod_{j\in\mathcal{J}}X_j$ is the product space of these spaces, then for any $\displaystyle x\in X$ we have that if $\displaystyle N$ is a neighborhood of $\displaystyle X$ then $\displaystyle \pi_j(N)$ is a neighborhood of $\displaystyle \pi_j(x)$ for each $\displaystyle j\in\mathcal{J}$

4. The converse is true if $\displaystyle \mathcal{J}$ is finite.

5. Using this we can show that if $\displaystyle \left\{X_j\right\}_{j\in\mathcal{J}}$ is a collection of topological spaces, and $\displaystyle \mathcal{D}_\ell$ is dense in $\displaystyle X_\ell$ for each $\displaystyle \ell\in \mathcal{J}$ then $\displaystyle \prod_{j\in\mathcal{J}}\mathcal{D}_j$ is dense in $\displaystyle X = \prod_{j\in\mathcal{J}}X_j$ with the product topology.

6. Consequently, if $\displaystyle X_1,\cdots,X_n$ are separable topological spaces then $\displaystyle X = X_1\times\cdots\times X_n$ 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.