Consider the following pushout in the category Top $:$
$$\require{AMScd} \begin{CD} A @>{f}>{}> B\\ @V{g}VV @VV{i_1}V\\ Y @>{i_2}>{}> X\end{CD}$$
Question $:$ Does it induce a pushout of the following form
$$\require{AMScd} \begin{CD} A \times I @>{f \times \text {id}}>{}> B \times I\\ @V{g \times \text {id}}VV @VV{i_1 \times \text {id}}V\\ Y \times I @>{i_2 \times \text {id}}>{}> X \times I \end{CD}$$
Let $i_1 \times \text {id} = i_1',i_2 \times \text {id} = i_2', f \times \text {id} = f'$ and $g \times \text {id} = g'.$ So I need to show the universality of the triple $(X \times I, i_1', i_2').$ That is given any topological space $Z$ and given any two maps $j_1 : B \times I \longrightarrow Z$ and $j_2 : Y \times I \longrightarrow Z$ with $j_1 f' = j_2 g'$ there exists a unique map $v : X \times I \longrightarrow Z$ such that $v\ i_1' = j_1$ and $v\ i_2' = j_2.$
So we know the following $:$
$(1)$ $i_1(f(a)) = i_2 (g(a)),$ for all $a \in A.$
$(2)$ $j_1 (f(a),t) = j_2 (g(a),t),$ for all $a \in A$ and for all $t \in I.$
With these two facts in mind we need to construct a unique $v : X \times I \longrightarrow Z$ such that $v(i_1(b),t) = j_1(b,t)$ and $v(i_2(y),t) = j_2 (y,t),$ for all $b \in B, y \in Y$ and $t \in I.$
How do I find such a $v\ $? Would anybody please help me in this regard? Thanks for reading.
Notice that for every $t\in I$, the maps $j_1,j_2$ induce a pair of maps $j^t_1:B\rightarrow Z$ and $j^t_2:Y\rightarrow Z$ as $$j^t_1(b)=j_1(b,t)\qquad j^t_2(y)=j_2(y,t)$$ The two maps $j^t_1,j^t_2$ satisfy $j^t_1\circ f=j^t_2\circ g$, so there is a unique continuous $\phi^t:X\rightarrow Z$ such that $\phi^t\circ i_n=j^t_n$, for $n=1,2$. Define $\phi:X\times I\rightarrow Z$ as $\phi(v,t)=\phi^t(v)$.
Otherwise, you can use the fact that $I$ is locally compact and Hausdorff, so the functor $I\times -:\mathbf{Top}\rightarrow\mathbf{Top}$ has a right adjoint $-^I:\mathbf{Top}\rightarrow\mathbf{Top}$ sending a space $B$ to the space $$B^I=\{f:I\rightarrow B:f\text{ continuous}\}$$ with the compact-open topology. In particular, if $I\times-$ is a left adjoint, it preserves colimits (and pushouts in particular).