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!
 A: 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.
A: Continuous image of a connected set is connected.
Continuous image of a complete set is not complete.
Continuity does not preserve Cauchy sequences, unless it's a uniform continuity.
A: *

*Connectedness and path connectedness

*If $f$ is a local homeomorphism, then $f$ is an open and closed map.

*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.

*Second-countability is preserved under open maps.

*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.
A: Following from muad's comment one could say that
1) $A \subseteq X_2$ is open, then $f^{-1}(A) \subseteq X_1$ is open 
2) same thing for closed sets
which I think are more simple conditions for characterizing a continuous function as opposed to those in your list, since the concept of open and closed sets are "simpler" than compact of connected sets.
