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Of late, I've been finding myself increasingly interested in questions of the form: "Suppose we change this definition in the following way. Does this change break anything?" For instance, suppose we change the definition of a metric space so that $(X,d)$ can be a metric space even when $d$ is defined for values outside $X^2$. So we can say... "Suppose $X \subseteq Y$ and $(Y,d)$ is a metric space. Then $(X,d)$ is a metric space." Would this break anything?

My question is, are questions of the form "would this break anything?" considered to be "real questions"? Is there perhaps a better website for this sort of thing?

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    $\begingroup$ I am not sure about this particular question, because any introduction to metric spaces states the fact that any subset is also a metric space in precisely this sense. // In general, it would be nice if a question indicated in which direction your thoughts are going. Remember the basic criterion: if someone reads your question and another user's reply to it, they should be able to tell whether the reply actually answers the question. $\endgroup$
    – user53153
    Commented Jan 25, 2013 at 5:53
  • $\begingroup$ Technically you have to restrict $d$ to $X$, forming a new metric $d|_X$. So $(X,d|_X)$ is a metric space, but $(X,d)$ maybe isn't. $\endgroup$
    – goblin GONE
    Commented Jan 25, 2013 at 5:56
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    $\begingroup$ I think the issue here is with the formalism. The notation $(Y,d)$ typically indicates that $d$ is defined on $Y\times Y$, so if this is the convention, there is abuse of notation in saying that $(X,d)$ is a metric space. Of course, some authors define things so, as long as the domain of $d$ contains $X\times X$, the notation $(X,d)$ is fine. But the point here is that whatever formalism is chosen, it is a convention, while the "real idea" is something else. $\endgroup$
    – Andrés E. Caicedo
    Commented Jan 25, 2013 at 5:58
  • $\begingroup$ What is a query that produces examples of these kinds of questions? I've seen many before and asked some myself but I'm having trouble finding more with a search. I think some examples could create a common ground for discussion. $\endgroup$
    – Dan Brumleve
    Commented Jan 25, 2013 at 6:06
  • $\begingroup$ A recent example: What would happen if we created a vector space over an integral domain/ring. (sic) $\endgroup$
    – user856
    Commented Jan 25, 2013 at 6:58
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    $\begingroup$ Jarrell: I thought you said to stay on the path! Old Man: Yes, but you must know when to break the rules! $\endgroup$
    – draks ...
    Commented Jan 25, 2013 at 21:21

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Not only are these real questions, but these are precisely the sort of questions you will find yourself asking more and more as your understanding of a subject deepens.

Perhaps talking of "breaking" anything is not quite how I would phrase it myself, but the point remains: To understand a subject, you need to understand how its components fit together, whether their connections are tenuous or withstand some modifications.

In particular, doing research this is actually done all the time: You attempt to modify an argument, and see where problems may appear, and how can they be fixed, and how far this process can go.

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  • $\begingroup$ I'm in total agreement. Is the soft-question tag appropriate in this sort of case? $\endgroup$
    – Dan Brumleve
    Commented Jan 25, 2013 at 5:54
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    $\begingroup$ @Dan, I think that tag is for questions with no significant mathematical content (I may be wrong about that), whereas the kind of question we are talking about can have lots of math content. $\endgroup$
    – Gerry Myerson
    Commented Jan 25, 2013 at 5:56
  • $\begingroup$ @Gerry the tag is described as being "for questions which admit no definitive answer" so perhaps its definition is too broad. $\endgroup$
    – Dan Brumleve
    Commented Jan 25, 2013 at 5:57
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    $\begingroup$ @Dan: "Would this break anything" admits definitive answers, in principle. The answer is either "Yes, it breaks some things" or "No, it doesn't break anything." $\endgroup$
    – Willie Wong Mod
    Commented Jan 25, 2013 at 9:00
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    $\begingroup$ @Willie, I think breakage is subjective. $\endgroup$
    – Dan Brumleve
    Commented Jan 25, 2013 at 9:15
  • $\begingroup$ @Dan, perhaps what the tag says $\ne$ how we actually use it. Have you looked at many questions with that tag, to get some sense of m.se practice? $\endgroup$
    – Gerry Myerson
    Commented Jan 25, 2013 at 11:40
  • $\begingroup$ @DanBrumleve At least it presents the question "does there exist something satisfying my new definition that does not satisfy the old definition?" $\endgroup$
    – Alexander Gruber Mod
    Commented Feb 1, 2013 at 14:00

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