Is every funky topology a topology?

Question

Whenever $X$ is a set, define that a funky topology on $X$ is a collection of subsets of $X$ deemed "open" such that:

• $X$ is open.
• If $A,B \subseteq X$ is open, then $A \cap B$ is open.
• If $\mathcal{C}$ is an open cover of $X$, then for all $A \subseteq X$, if $$\forall U \in \mathcal{C}(A \cap U \mbox{ is open}),$$ then $A$ is open.

It's clear that every topology is a funky topology.

Question. Does the converse hold?

Answer 1

No. For instance, $\{X\}$ is a funky topology on any set $X$, but is not a topology unless $X$ is empty.

It also does not suffice to assume the empty set is open. For instance, if $X=\{a,b,c\}$, then $\{\emptyset,\{a\},\{b\},X\}$ is a funky topology but not a topology.