Wikipedia:Reference desk/Archives/Mathematics/2013 February 5

= February 5 =

A quetion in the real line topology
Hi! Here is my question: Given a countable union of countable sets $$ A_i $$ such that $$ \cup A_i = R $$. Does it implies that at list one of this sets is dense in some open segment of R? and if yes, how to prove it? Thanks! Topologia clalit (talk) 12:32, 5 February 2013 (UTC)
 * Are you sure you mean a countable union of countable sets? Because that requires rejecting the axiom of choice.--190.18.159.129 (talk) 12:49, 5 February 2013 (UTC)
 * Nevermind, our Baire category theorem article claims it can be proven for the reals without using choice. So the answer is yes, it follows from the Baire category theorem.--190.18.159.129 (talk) 13:08, 5 February 2013 (UTC)

Hi, As a matther of fact I have looked over this theorem. But I don't se how it's connected. The written states that the reals are a Baire space while: "A Baire space is a topological space with the following property: for each countable collection of open dense sets Un, their intersection ∩ Un is dense." But in my question there is a countable union of sets (not all countable) where $$ \cup A_i = R $$. But it is not given that the sets are dense in R. and I want to show that at least one of them is dense in some open interval of R. I don't need to show that their intersection is dense.. I don't understand the connection.. Thanks Topologia clalit (talk) 13:58, 10 February 2013 (UTC)
 * The connection is via the complement. Let $$B_i$$ denote the closure of $$A_i$$.  You wish to show that one of the $$B_i$$ contains an interval.  Since $$\bigcup_i B_i \supseteq \bigcup_i A_i = \mathbb{R}$$, it follows that $$\bigcap_i (\mathbb{R} - B_i) = \emptyset$$.  But each $$(\mathbb{R} - B_i)$$ is an open set, so by the contrapositive of the Baire category theorem, at least one of the $$ (\mathbb{R} - B_i)$$ is not dense.  That means there is some open interval it does not intersect, and thus $$B_i$$ contains that open interval.--190.18.159.129 (talk) 03:49, 11 February 2013 (UTC)