I wondering if you guys could help me figure out some stuff. Let $(X,\tau_{1})$ and $(Y,\tau_{2})$ be topological spaces.
I know that:
if $(X\wedge Y)\in T_{0}\implies X\times Y \in T_{0}$
if $(X\wedge Y)\in T_{1}\implies X\times Y \in T_{1}$
if $(X\wedge Y)\in T_{2}\implies X\times Y \in T_{2}$
if $(X\wedge Y)\in T_{3}\implies X\times Y \in T_{3}$
$T_{3}$ meaning that $\forall x \in X\quad \wedge\quad \forall A^{closed}\subset X \quad \exists W,V \text{ such that } W\cap V =\emptyset\quad $ and W and V are neighbourhoods for x and A.
I am wondering if the verse can also be said. That is if $X\times Y \in T_{i}\implies (X\wedge Y)\in T_{i}$?
In other words can we replace the implication with equivalence? And if not when can do that.
Thanks for any help
Edit:What about compactness and connectedness?
First remark: $X\land Y \in T_0$ makes no sense. You probably mean (if you really want to use symbolism for this at all): $(X\in T_0) \land (Y \in T_0)$ instead.
If $p \in X$, the subspace $\{p\} \times Y$ is homeomorphic to $Y$ and if $q \in Y$, $X \times \{q\}$ is homeomorphic to $X$.
And all these separation properties $T_0, \ldots T_3$ are hereditary.
Compactness and connectedness are preserved by continuous images so there use that $\pi_X[X \times Y] = X$ and projections are by definition continuous in the product topology.