Is there a natural topology on the set of open sets?

For example, suppose you want to show R (the reals) is uncountable. One famous proof is based on Cantor's diagonal reasoning. But to use it you must know every real number has a decimal representation, actually you have to work not on R but on the field...

The interior Ǻ of a set A is the union of all open sets included in A a) prove that, for any set A, its interior Ǻ is open. b) prove that, for any set A, we have Ǻ ⊆ L(A) Please explain :)

please think on the lines of function f to be defined on each set such that it is complex valued and has k continuous differentials.

A is pre open iff A subset of interior of A clouser

Help me plan my (somewhat pathetic) workouts! I've been on the hack diet for nine months, and have lost lots of weight (65 down, have another 15 or so to go). I also started doing the exercise program recommended in the diet, though I've substituted...

If you encounter the Microsoft Lync 2013 Installation topology error, here is your fix. Installing Lync Server 2013: Topology File Share Error Fix Today, the FortressITX engineers came across an issue with Microsoft Lync 2013. While installing the program...

By "train your model iteratively across several training/validation sets," I mean split original training set into several data sets, train on remaining training records, predict on validation set 1, append the predicted values to validation...

I read that the set of all sets is a contradictory concept. I thought about this. If U is such set, then U must be an element of U (not only that U is - of course- a subset of U, but that U belongs to U like 2 belongs the reals). Though this sounds kinda...

Sets of equal cardinality aren't necessarily themselves equal, so that it is possible to construct two sets of equal cardinality with one having an infinite number of elements more than the other. Can one reasonably construct an argument that makes a...