Talk:Point Finite Set of Open Sets in Separable Space is Countable

Explanation
The explanation is a basic fact about sets, and has nothing to do with topology:

Suppose that:

$\mathcal F$ is a set of sets,

$S$ is a set, and

Every element of $\mathcal F$ contains an element of $S$ (that is, $\forall V \in \mathcal F: \exists x: x \in V \cap S$).

Then $\mathcal F = \bigcup_{x \in S} \{ V \in \mathcal F: x \in V \}$.

This can be proved just from the definition of union and such, but it hardly seems to belong in the middle of a topological proof, and it's not so easy to come up with a name for this particular triviality.

As for denseness, another characterization is that every non-empty open set in the space contains an element of the dense set. --Dfeuer (talk) 21:08, 10 May 2013 (UTC)


 * Oh okay, you've done it. --prime mover (talk) 22:18, 10 May 2013 (UTC)

name change
Point Finite is defined on this site only for covers. This is as defined on Steen & Seebach. Either:
 * a) The concept of point finiteness is appropriate only for covers, in which case this entry needs to be named back again to what it was and returned to being for covers and not general sets of sets;

or
 * b) The definition of Point Finite needs to be expanded to the more general Set of Sets.

If b) then, on the Point Finite page, a printed source is to be cited and some explanatory words need to be put in place so as to clarify for stupid doofuses (i.e. everyone who is not perfectly au fait and fluent with the minute details of topology) that the concept is treated differently by different sources.

If there are no sources that you can find which use this more general version of Point Finite, i.e. it's an idea of your own, then stow it onto your own page and return this back to whence it came. --prime mover (talk) 21:26, 11 May 2013 (UTC)


 * Choice (b). Kelley chapter 5 problem V. --Dfeuer (talk) 23:22, 11 May 2013 (UTC)


 * In that case you have work to do. You cannot leave this page with the definition compromised like it is - please sort out that page to match. --prime mover (talk) 07:52, 12 May 2013 (UTC)