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It can be shown that if $f$ is a map of sets, $f:X\rightarrow Y$ say, that for $A_\lambda \subseteq X, \lambda \in \Lambda$, an indexing set, then:

$$f\left(\bigcup\limits_{\lambda}A_\lambda\right)=\bigcup\limits_{\lambda}f\left(A_\lambda\right)$$

and

$$f\left(\bigcap\limits_{\lambda}A_\lambda\right)\subseteq\bigcap\limits_{\lambda}f\left(A_\lambda\right).$$

I was just wondering are there functions such that equality holds for the second statement for all collections of subsets $\{A_\lambda\}_{\lambda \in \Lambda}$ of $X$ where $\bigcap\limits_{\lambda}A_\lambda \neq \emptyset$?

I would suppose constant maps to be an example. Any others? I'm just wondering if there might be a condition for it.

This seemed relevant but it relies on the sets having topologies on them, with "nice" properties.

Continuous image of the intersection of decreasing sets in a compact space

I was thinking about this in relation to a mapping of closed sets in a basis for a topology, and I suppose using topology would also be relevant then.

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snulty
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    The equality in second statement is a characterization of injectivity. – leo May 01 '14 at 20:23
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    I am sure, about six hundred thousand to the power of Graham's number's iteration of the Ackermann function whose input is the value of the Busy Beaver function at Googleplex. This is a duplicate. – Asaf Karagila May 01 '14 at 20:26
  • @AsafKaragila how would one go about finding the original non-duplicate question? – snulty May 01 '14 at 20:28
  • Not sure about "original" but there is a thread with a list of results about preimages, images and Boolean operations. I'd start there. Searching in [tag:elementary-set-theory] with the expression is:q wiki:1 might be a good place for start. – Asaf Karagila May 01 '14 at 20:33
  • @leo does it not work for constant maps? – snulty May 01 '14 at 20:35
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    More concretely: http://math.stackexchange.com/questions/359693/overview-of-basic-results-about-images-and-preimages is the thread I had in mind from where you should start your quest for fire. I mean quest for duplicate. – Asaf Karagila May 01 '14 at 20:36
  • @AsafKaragila I'm looking at that now, thanks. – snulty May 01 '14 at 20:38
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    If you only allow families with nonempty intersection then yes, it works for constant functions, but then it no longer is a characterization of injectivity. – leo May 01 '14 at 20:54
  • @leo ok, but injectivity wasn't the sole purpose of this question – snulty May 02 '14 at 12:05
  • @leo The question would be are there any others, injective functions, constant maps and...? or no more. I appreciate you pointing out injective maps though – snulty May 02 '14 at 12:15

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