My professor sort of skimmed through this concept, giving only the definition and one example (i.e. $(0,1) \subset \mathbb{R}$ vs. $(0,1) \subset \mathbb{R}^2$, where $(0,1)$ is open relative to $\mathbb{R}$ but not $\mathbb{R}^2$), but no real explanation. I understand, more or less, this particular example, but I'm having trouble understanding this more generally.
Can somebody please intuitively explain this concept in the context of a metric space? (If it matters, we are using Rudin's Principles Of Mathematical Analysis).
Best Answer
In order to know what "open relative to" means, you have to first know what "open" means:
Now, if we're working in some metric space and consider a subset A of that space, you can, if you wish, disregard the rest of the space and think of A as a metric space in itself. The notion of "open relative to A" is what you get when you realize, looking at the above definition, that some subsets of A that are not open subsets of the larger space may still be open subsets of A.
For example, suppose you look at the interval I=[0,1] as a metric space in itself, and disregard the rest of the real line. Then the subset S=(0.5,1] may seem at first not to be an open set, because you think, "Hey, there are points arbitrarily close to 1 that are not in S, so 1 is actually a boundary point!" Well, yes... except that none of those arbitrarily-close points are actually in I, so from the point of view of I as a space in itself, they don't exist. Every point in I sufficiently close to 1 is also in S, so if we want to work in I alone, we have to concede that S is open.
The moral of this story is that sets aren't inherently open: the definition depends on what metric space they are considered as subsets of. If you are only talking about one space it's harmless to just write "open," but otherwise you may have to be more specific. If you are considering both $\mathbb{R}$ and I, for example, writing "(0.5,1] is open" is ambiguous. So instead you write "(0.5,1]" is open relative to I."