  We consider certain infinite dimensional modules of level 1 for the double
Yangian $\text{DY}(\mathfrak{gl}_2)$ which are based on the Iohara-Kohno
realization. We show that they possess topological bases of
Feigin-Stoyanovsky-type, i.e. the bases expressed in terms of semi-infinite
monomials of certain integrable operators which stabilize and satisfy the
difference two condition. Finally, we give some applications of these bases to
the representation theory of the corresponding quantum affine vertex algebra.
