Currently, as far as I can tell, when one opens a question and it gets closed after, there is no dynamic notification about, and instead it will only be noticed after a refresh, or if one tries to post an answer (or a comment if the question is deleted). This can be very inconvenient, as more than once I have researched and worked out an answer for a question only to be found the moment I try to post it that it was closed a while ago.
There are dynamic notifications when a post is edited, which is very useful, and new comments appear on under posts immediately after they are posted, so I don't see why closing the question should be any different.
Note this is not the same question as Dynamically update close vote counts, although I think dynamically updating close votes could also be useful, for example if you see a question is one vote away to be closed, you may want to reconsider if it is worth continue researching it, as it is likely that it will be closed before you can post an answer. Different users may have different views on how to use that information, but the point is that I don't think it makes sense for different users to have different information about a question simply because they opened it a bit sooner or later.
EDIT: For clarity, gif showing the fact.