TITLE: Topological properties preserved by continuous maps
QUESTION [26 upvotes]: 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!

REPLY [3 votes]: Continues image of a connected set is connected.
Continues image of a complete set is not complete. 
Continuity does not preserve Cauchy sequences, unless it's a uniform continuity.