Let $X$ and $X'$ denote a single set in the topologies $T$ and $T'$ respectively; Let $Y$ and $Y'$ denote a single set in the topologies $U$ and $U'$ respectively.
Assume the sets are non empty.If the product topology on $X' \times Y'$ is finer than the product topology on $X \times Y$ ,does it imply that $T'$ contains $T$ and $U'$ contains $U$?
My attempt is,we know that the collection of $B \times C$ (where $B$ is a basis element of $T$ and $C$ is a basis element of $U$) forms a basis for the product topology on $X \times Y$ and similarly for $X' \times Y'$ we can do the same.
So,if $B$ is a basis element of $T$ and $x$ belongs to $B$,then $(x,y)$ belongs to $B \times C$ for some $y$ in $C$(where $C$ is a basis element of $U$).Then there exists $B' \times C'$(where $B'$ is a basis element of $T'$ and $C'$ is a basis element of $U'$) such that $(x,y)$ belongs to $B' \times C'$ which is contained in $B \times C$.Thus $x$ belongs to $B'$ which is contained in $B$.similarly we can do this to show $U'$ contains $U$.
Is this correct?