A question about the hierarchy of topologies on a given set

Question

It's easier to understand with examples:

• Every finer topology than a Hausdorff topology is hausdorff.

• Every coarser topology than a compact topology is compact.

What are the full set of properties which admit a statement of this kind?

Answer 1

A connected set remains connected in a coarser topology.

The boundary of a subset increases with coarser topologies and decreases with finer topologies.

Continuity is preserved by coarsening the image or refining the domain.