Definition talk:Open Set/Metric Space

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This page is confusing because of the non-standard use of "neighborhood" to mean "open ball". A neighborhood of a point is just any set containing an open set containing that point.

In a general topological space maybe, but not from the sources I can find on a metric space. Where's your sources? Quote them. --prime mover 17:03, 27 October 2011 (CDT)


Question

There seems to be a deliberate attempt to use different notations on different pages. I'm wondering what's the reason behind all that. For example, in the definition of set union, we write:

$\displaystyle \forall x \in \bigcup \Bbb S: \exists S \in \Bbb S: x \in S$

On this site, I haven't ever seen something like this:

$\displaystyle \forall x \in \bigcup \Bbb S: \exists S \left({x}\right) \in \Bbb S: x \in S \left({x}\right)$

For me, writing "$S \left({x}\right)$" suggests that we are talking about some mapping $\displaystyle S: \bigcup \Bbb S \to \Bbb S$. Correct me if I'm wrong.

In any case, there seems to be a desire to use the latter notation on this page (and perhaps on several other isolated pages). Why? I don't see it on most pages, and it doesn't add any meaning. --abcxyz (talk) 21:05, 19 October 2012 (UTC)

It's there for the ignorant and uninitiated, whom are not capable (yet) of noticing that $\epsilon$ may depend on $y$. Any soul acquainted with quantifiers shouldn't be prone to the mistake of believing otherwise. I vote for this horror to be eliminated. --Lord_Farin (talk) 21:30, 19 October 2012 (UTC)
Why can't $\epsilon(y)$ mean "a value $\epsilon$, which is dependent upon (and can be considered to be a function of) $y$"? --prime mover (talk) 21:40, 19 October 2012 (UTC)
For one, $\epsilon(y)$ is not uniquely determined by $y$, so technically this isn't a function. I am more used to subscripts to denote such dependencies (and even then, only upon introducing, e.g. putting $\epsilon = \epsilon_y$ would be admissible to me); would the vote turn against me, I could settle for $\epsilon = \epsilon(y)$, I think. But I'm definitely against writing $\epsilon(y)$ always (what if we want to use the same $\epsilon$ multiple times? It's just awkward. --Lord_Farin (talk) 21:48, 19 October 2012 (UTC)
Meh. --prime mover (talk) 22:37, 19 October 2012 (UTC)