$\tau_1,\tau_2,\tau_3$ are topologies on a set such that $\tau_1\subset \tau_2\subset \tau_3$ and $(X,\tau_2)$ is a compact Hausdorff space. Could any one tell me which of the following are correct?
- $\tau_1=\tau_2$ if $(X,\tau_1)$ is compact Hausdorff.
- $\tau_1=\tau_2$ if $(X,\tau_1)$ is compact.
- $\tau_2=\tau_3$ if $(X,\tau_3)$ is Hausdorff.
- $\tau_2=\tau_3$ if $(X,\tau_3)$ is compact.