Theorem 16.3.
If $A$ is a subspace of $X$ and $B$ is a subspace of $Y$ , then the product topology on $A × B$ is the same as the topology $A × B$ inherits as a subspace of $X × Y$ .
While readin Topology 2ed, J. Munkres I had read above Therorem. However, can't understand what "inherits as a subspace".
Which mathematical object does inherit refer to in this context?
On
$A \times B$ is a subspace of the topological space $X \times Y$, so you can apply the definition of "subspace topology" to define a topology on $A \times B$.
On
This means that a subset $O$ of $A\times B$ is open if and only if thereis an open subset $O^\star$ of $X\times Y$ such that $O=O^\star\cap(A\times B)$.
On
$X \times Y$ has the product topology w.r.t. the topologies of $X$ and $Y$.
$A \times B$ is a subspace of $X \times Y$, so it can be given a topology $\mathcal{T}_1$ as the subspace topology of the product topology on $X \times Y$.
On the other hand, $A$ and $B$ are subspaces of $X$ resp. $Y$ in their respective subspace topologies. So they can be seen as their own space. And as such $A \times B$ can be given a topology $\mathcal{T}_2$: the product topology w.r.t. the (subspace) topologies of $A$ and $B$.
The problem is: show $\mathcal{T}_1 = \mathcal{T}_2$.
The "topology $A\times B$ inherits as subspace of $X\times Y$" is the subspace topology of $A\times B$ when you see it as a subspace of $X\times Y$ via the natural inclusion map $$i:A\times B\longrightarrow X\times Y$$ given simply by $i(a,b) = (a,b)$.