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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!

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    Wouldn't it be better to start with inverse images rather than images?2010-08-26
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    There are a couple here: http://en.wikipedia.org/wiki/Continuous_function#Useful_properties_of_continuous_maps - Connected and Path-Connected2010-08-26
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    $f(\text{Hausdorff}) \neq \ \text{Hausdorff}$.2010-08-26
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    $f(\text{locally (path) connected}) \neq \ \text{locally (path) connected}$ (but quotients of locally (path) connected spaces are locally (path) connected). $f(\text{locally compact}) \neq \ \text{locally compact}$.2010-08-26
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    I just saw this question by coincidence: on page 510 of Engelking's *General Topology* there's a rather extensive table called *Invariants and inverse invariants of mappings*. You'll find topological properties with indication of whether they are preserved by (various kinds of) continuous maps or not (such as open maps, closed maps, quotient maps, perfect maps, etc.). For mere continuous most things have been mentioned: simple covering properties (variations on compactness, connectedness, Lindelöf) and separability. More complicated covering properties such as paracompactness aren't preserved.2012-06-06

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