Topological properties preserved by continuous maps

A continuous function does not always map open sets to open sets, but a continuous function will map compact sets to compact sets. One could make list of such preservations of topological properties by a continuous function $f$: $$ f( \mathrm{open} ) \neq \mathrm{open} \;,$$ $$ f( \mathrm{closed} ) \neq \mathrm{closed} \;,$$ $$ f( \mathrm{compact} ) = \mathrm{compact} \;,$$ $$ f( \mathrm{convergent \; sequence} ) = \mathrm{convergent \; sequence} \;.$$ Could you please help in extending this list? (And correct the above if I've erred!)

Edit. Thanks for the several comments and answers extending my list. I was hoping that I could see some common theme among the properties preserved by a continuous mapping, separating those that are not preserved. But I don't see such a pattern. If anyone does, I'd appreciate a remark. Thanks!

Solution 1:

1. Connectedness and path connectedness
2. If $f$ is a local homeomorphism, then $f$ is an open and closed map.
3. If $f$ is onto and the domain is normal (can separate closed sets) or the map has compact fibers, then the image will be Hausdorff if the domain is.
4. Second-countability is preserved under open maps.
5. The image of a simply connected space need not be simply connected.

That's off the top of my head.. there are many more. Also, this should probably be community wiki.

Solution 2:

These examples may be silly, but just to add to your list:

1) Sequential compactness (i.e. every sequence has a convergent subsequence)

2) Countable compactness (i.e. every countable open cover has a finite subcover)

3) $\sigma$-compactness (i.e. the space is a countable union of compact sets)

Less trivially, as mentioned in the Wikipedia article, the Lindelof property and separability are both preserved.