Is the collection $\tau_\infty = \{U:X-U$ is infinite or empty or all of $X\}$ a topology on $X$?

Can someone please verify my proof?

Is the collection $\tau_\infty = \{U:X-U$ is infinite or empty or all of $X\}$ a topology on $X$?

No. Let $X = \mathbb{R}$.

Clearly, $\{x\} \in \tau_\infty$ for all $x \in X$.

Set $\displaystyle{B = \cup_{x \neq 0} \{x\}}$. Clearly, $X - B = \{0\}$, but $\{0\} \notin \tau_\infty$. Since $\tau_\infty$ is not closed under arbitrary unions, it does not define a topology on $X$.

Correction: you shall say but $\{0\}$ is not infinite, so $B\notin \tau_\infty$. Otherwise the proof is correct. Your claim that $\{0\}\notin \tau_\infty$ is contradictory, and not the one you want.