This is a problem 1 from Munkres pg 83. I'm trying to solve for self study. Let X be a topological space; let A be a subset of X. Suppose that for each x belonging in A there is an open set U containing x such that U is a subset of A. Show that A is open in X. I'm not sure exactly how an open subset of A makes the whole A open. Thanks
By definition, X and the empty set are open in X. Also, an arbitrary union of open sets is open, and a finite intersection of open sets is open.
re This is what I came up with, Given an open set U sub n (where n belongs to J) containing x, then A = The union of U sub n (n belongs to J) , by definition "an arbitrary union of open sets is open" hence A is open, would this be ok, thanks
So for each x in A, there exists an open set U(x) such that [itex]x \in U(x) \subseteq A[/itex]. Let O be the union of all such open sets, [itex]O = \bigcup_{x \in A} U(x)[/itex]. Show that O is open, and that O = A. (Remember, the union of any collection of open sets is open, by definition).
waht, you seem to need to practice reading definitions and using them directly. this problem is almost completely trivial logically, so not getting it may mean you have a gap in understanding doing proofs. you might want to consult some elementary proof books, like An introduction to mathematical thinking, or Principles of mathematics, or even the logic chapter of Harold Jacobs excellent high school book Geometry, since many high schools no longer offer such fine geometry training including proofs. I am also puzzled that you find yourself in a course like Munkres without this training. Does your school offer a proofs course which you may have missed? If so, you might try that first. Or do they just think plunging right into Munkres is sufficient practice in making proofs? You really need basic rules of mathematical logic first. best wishes.
re I would love to take a topology course since I find it so fascinating but it is not going to happen. So I purchased a copy of Munkres based on excellent reviews and I'm studying it at my spare time. My math background is past diffrential equations and linear algebra, but have been studying other subjects when time allowed. At the beginning, such elementary proof seemed impossible but I'm starting to connect the dots. I still don't get even 1% of the book heh. Thanks for the help