Show that To is productiveSolutionLet xy be a point in XxY L

Show that: T_o is productive

Solution

Let (x,y) be a point in XxY.

Let (a,b) be in XxY , distinct from (x,y). So both x=a and y=b cant hold.

Let x , a be distinct, say.

SInce X has T₀ , there exists a neighbourhood U of x which does not contain a.

Consider UxY This is a neighbourhood of (x,y) (in the product topology) which does not contain the point (a,b ), as a is not in U.