Q1:-
If ( X , T) & ( Y , J ) are both Hausdrff space`s , so is ( X x Y , P X x Y ) .
remark:- P X x Y = the topology of the basis of ( X x Y )
Q2:-
A subspace of a Hausdrff space is also a Hausdrff space .
Whith many many thnx
Q1:-
If ( X , T) & ( Y , J ) are both Hausdrff space`s , so is ( X x Y , P X x Y ) .
remark:- P X x Y = the topology of the basis of ( X x Y )
Q2:-
A subspace of a Hausdrff space is also a Hausdrff space .
Whith many many thnx
If $\displaystyle (x_1,y_1),(x_2,y_2)\in X\times Y$ pick open sets $\displaystyle U_1,U_2\subset X$ and $\displaystyle V_1,V_2\subset Y$ (all open in their respective topologies) such that $\displaystyle x_i\in U_i$ and $\displaystyle y_i\in V_i$ and $\displaystyle U_1\cap U_2=\emptyset = V_1\cap V_2$. What happens if $\displaystyle (U_1\times V_1) \cap (U_2\times V_2)\neq \emptyset$?
Let $\displaystyle x,y\in Y\subset X$ pick $\displaystyle U,V \subset X$ such that $\displaystyle x\in U$, $\displaystyle y\in V$ and $\displaystyle U\cap V= \emptyset$ then what about $\displaystyle U\cap Y$ and $\displaystyle V\cap Y$
first of all :- many many thnk`s for your help
now
the first section is 100% true
but then ....???
Let $\displaystyle x,y\in Y\subset X$ pick $\displaystyle U,V \subset X$ such that $\displaystyle x\in U$, $\displaystyle y\in V$ and $\displaystyle U\cap V= \emptyset$ then what about $\displaystyle U\cap Y$ and $\displaystyle V\cap Y$[/QUOTE]
how came Y be a subset of X ??!!!!!
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now i think that , we begin with
let $\displaystyle (U_1\times V_1) \cap (U_2\times V_2)\ = h $
$\displaystyle h \in (U_1\times V_1) \ $ & $\displaystyle h \in (U_2\times V_2)\ $
then same how we will get a contradiction ( how i don`t know ) ???