Given a topological space $(X,\mathcal{O})$ can one assign a natural topology to $\mathcal{O}$ such that

1) The intersection of a compact set of open sets is again open,

2) The maps $\cap,\cup:\mathcal{O}^2\rightarrow \mathcal{O}$ are continuous,

3) For any continuous map $f: (X,\mathcal{O})\rightarrow (X',\mathcal{O}')$ is the induced map $f^{-1}: \mathcal{O}'\rightarrow \mathcal{O}$ also continuous?

Since these properties would allow taking the discrete topology and I am looking for a more interesting one let me also add:

4) For $X=\mathbb{R}$ the subset $\{]a,b[|a\le b\}$ of intervals has the topology coming from $\{(a,b)\in \mathbb{R}^2| a\le b\}/\Delta \mathbb{R}$.