Question. Are there contradictions with the following topology $\tau$? If not, is it an established topology known with a conventional name? Thanks in advance.
Suppose $(Y, <)$ is a non-empty complete lattice. That is, every non-empty subset has both a meet and a join. Equip $Y$ with a topology $\tau$, such that every converging net that is also order-preserving, w.r.t. its index directed set, converges to its join. That is for a directed set $(I, <=')$, if the net $(y_i)_{i\in I}$ converges under topology $\tau$, and satisfies $i_1<='i_2\Rightarrow y_{i_1}<=y_{i_2}$, then $y_i\rightarrow \bigvee y_i$ .