# Tag for semicontinuity?

A few days ago the tag was created and later removed. This lead to a discussion about this tag in the tag management thread. As suggested by Asaf, it might be more reasonable to discuss this in a separate question. (Among other reasons, there is more then one possible outcome of the discussion.)

The natural question concerning tagging questions about semicontinuous functions are:

• Should we have a separate tag for semicontinuity?
• If yes, should both lower and upper semicontinuity have separate tags or should they be under the same tag.
• If not, should we expand the scope of the tag to include this topic. (After all, some users have been using the tag in this way. But it would be nice to know the consensus of the community about this. And if the consensus is that the questions about semicontinuity belong here, this should be mentioned explicitly in the tag info.)
• Should the tag include also semicontinuity of multifunctions?

You can see a short discussion in chat between me and the user who created (upper-semicontinuity). And also that the tag-creator would prefer to have separate tags for lower and upper semicontinuous tags.

• Hi, I am the tag creator. As you can see, I avoid participation in meta (which is a corollary of my inner hermit social position :-) ) and instead of discussing the fate of the tag I prefer to answer a next question (in particular, the question which suggested me to create the tag still remains unanswered. :-) ) – Alex Ravsky Nov 23 '17 at 4:45
• Concerning the tag, I don’t insist on it, so feel free to deal with it as you wish. Proposing it I was guided by a general idea “Ordnung muss sein” :-) I can only add two small arguments about it. The presence of a tag allows interested people to simply track new relevant activity instead of searching the old that, or, conversely, ignore questions tagged by it if somebody is not interested in it (for instance, I have a list of more than two hundreds ignored tags concerning the topics in which I am lost). – Alex Ravsky Nov 23 '17 at 4:45
• Another point, which I already mentioned in chat is that notions of upper and lower semicontinuity for set-valued maps are essentially different. – Alex Ravsky Nov 23 '17 at 4:45
• Based on the voting and comments posted here I have accepted Michael Greinecker's answer. The tag (semicontinuous-functions) was created recently, the tag-info was created also based on the discussion here. (In particular, it includes that the tag is for semicontinuity of functions, but not for multifunctions.) @AlexRavsky Since you actually initiated created of the new tag, I thought that it would be polite to let you know about this, too. – Martin Sleziak Dec 6 '17 at 14:41

I think there should be tag that covers both upper-semicontinuous and lower-semicontinuous functions but excludes multi-valued functions. Upper- and semicontinuous functions are dual to each other and many duplicates of one version will be in terms of the dual concept.

I prefer to because the latter concept seems to include multi-valued functions. But if we treat a function as a special case of a multi-valued functions, all semi-continuity notions for multi-valued-functions boil down to ordinary continuity (but not upper-or semi-continuity). For multi-valued functions, a tag , an often used synonym in this context, could be used, but we don't have that many questions on it under now.

• Perhaps semicontinuous-functions AND semicontinuous-multifunctions AND semicontinuous-decompositions, although I suspect this last one is a bit too narrow for current topics of interest. Regardless, I think it would be useful to separate out basic real analysis type questions from the multifunction stuff. It also occurs to me that it might be better to use "semicontinuous" (alone or with other words) than "semicontinuity", to assist non-native English speakers, since the usage one almost always sees in definitions and theorems is "semicontinuous", and rarely "semicontinuity". – Dave L. Renfro Nov 26 '17 at 14:10

Here I mostly repeat the stuff I already said in the tag management thread.

• I think that it might be useful to have a tag for semicontinuoity of functions and multifunctions. (It seems to be a rather common topic in real analysis and general topology. This class of semicontinuous functions is useful in some contexts.)
• I think that it would be better to have single tag rather than two separate tags. (The two classes of functions are rather close to each other. Moreover, having too specific tags is not ideal since for a question we can have at most 5 tags.)
• The new tag could serve for questions about semicontinuity of both functions and multifunctions. (Notice that some people use the name hemicontinuity for the latter, and we have some questions using this name. As far as I can tell the term semicontinuity is also used for multifuctions. For example Engelking's book, which I consider a classical reference, uses the name semicontinuity for set-valued mappings; see Problem 1.7.17.)
• The problem with using the same tag for semicontinuous functions and semicontinuous multifunctions is that the terminology is inconsistent here. If we identify a function with a multifunction that happens to have singletons as values, the function is upper-semicontinuous as a multifunction iff it is lower-semicontinuous as a multifunction iff it is continuous as a function. Indeed, that is the reason why many people (including me) prefer to talk about hemicontinuity in the case of multifunctions. – Michael Greinecker Nov 22 '17 at 18:22
• I agree with you that it's favourable to create a single semicontinuity tag. Also there's no need to create separate "upper/lower SC" tags because it's just a matter of the sign, at least in the real-valued case. I don't know much about hemicontinuity, but I guess it won't have many related posts anyway. – Vim Nov 22 '17 at 18:22
• @MichaelGreinecker As I said, I am pretty sure that both names are commonly used for multifunctions. Your suggestion would be to have a separate (hemicontinuity) tag - which would be for multi-valued functions? – Martin Sleziak Nov 22 '17 at 18:30
• @MartinSleziak Yes, I would have a tag for semicontinuous functions, but not multifunctions. I don't know whether we need to have a hemicontinuity tag for multifunctions, but I don't think we should have an umbrella talk for clearly distinct concepts. It would be like having a "regular" tag that covers every use of regularin mathematics. – Michael Greinecker Nov 22 '17 at 19:02
• @MichaelGreinecker It is a valid point. (Although I think that using the same tag for corresponding notions for functions and multifunctions is much less extreme than you example with "regular".) Why not posting a separate answer suggesting a tag that would exclude multifunctions. (So that it is possible to vote on that suggestion and see what other users thing about these two suggestion.) – Martin Sleziak Nov 22 '17 at 19:30
• @MartinSleziak I interpreted your suggestion of having a tag for"semicontinuous functions" to already exclude multifunction. The regular case is of course extreme, but it was my impression that similarly named but distinct concepts should have distinct tags. – Michael Greinecker Nov 22 '17 at 19:33
• @MichaelGreinecker Since this is getting a bit too long, why don't we continue the discussion in chat. – Martin Sleziak Nov 22 '17 at 19:35
• @Michael: How about hemi-semi-demi-continuity, then? And if demicontinuity does not exist, we can define it! :P – Asaf Karagila Nov 22 '17 at 21:07
• @Asaf It's already too late. – Michael Greinecker Nov 22 '17 at 21:23
• @Michael: All the more reason! – Asaf Karagila Nov 22 '17 at 21:24