On an additive modification of the $\gamma$-property

For Tikhonov spaces, a sequence $(\gamma'_{k})_{k<\omega}$ of topological properties is defined, each of which is not stronger than the classical Gerlich–Nagy property ($\gamma$-property), and $\gamma'_{k +1}$ follows from $\gamma'_{k}$. The behavior of the index k under standard topological operations is studied. As one of the main results, it was established that, in contrast to the $\gamma$-property, taking a topological sum does not take the sequence $(\gamma'_{k})_{k<\omega}$ outside the sequence, but only leads to addition indices. In addition, the connection of the sequence $(\gamma'_{k})_{k<\omega}$ with the Lindelof property was found, as well as some other facts.