1. separation axioms

GIVE AN EXAMPLE :
X:normal
Y:normal
BUT X X Y : not normal
normal is not productive

2. Originally Posted by flower3
GIVE AN EXAMPLE :
X:normal
Y:normal
BUT X X Y : not normal
normal is not productive
The classic example (given in Kelley's General Topology, and attributed by him to Dieudonné and Morse, independently) uses ordinal spaces. Let $\Omega_0$ be the set of all ordinal numbers less than the first uncountable ordinal $\Omega$ and let $\Omega' = \Omega_0\cup\{\Omega\}$, each with the order topology. Then $\Omega_0$ and $\Omega'$ are both normal, but $\Omega_0\times\Omega'$ is not. (Let $A = \{(x,x):x\in\Omega_0\}$ and $B = \Omega_0\times\{\Omega\}$. Then A and B are closed and disjoint in $\Omega_0\times\Omega'$ but do not have disjoint open neighbourhoods. The proof is not that easy. You would do best to look it up in Kelley's book.)

3. Originally Posted by flower3
GIVE AN EXAMPLE :
X:normal
Y:normal
BUT X X Y : not normal
normal is not productive
Another example is a Sorgenfrey plane $\mathbb{R}_l^{2}$.

Even though a Sorgenfrey line is normal, the Sorgenfrey plane is not normal.
Let $\Delta = \{(x, -x) | x \in R \}$. As shown by the above link, no disjoint open sets can separate the disjoint closed sets $K= \{(x, -x) | x \in \mathbb{Q} \}$ and $\Delta \setminus K$ in $\mathbb{R}_l^{2}$. Intuitively, unions of the open rectangles containing $K$ and unions of the open rectangles containing $\Delta \setminus K$ in the above link somehow overlaps all the time. A rigorous proof needs more elaboration.

In contrast, $\mathbb{Q}$ and $\mathbb{R} \setminus \mathbb{Q}$ are disjoint closed sets in a Sorgenfrey line $\mathbb{R}_l$, which can be separated by disjoint open sets in $\mathbb{R}_l$ (they are both clopen sets).

thank you>>>