Let $K$ denote the set of all numbers of the form $1/n$ where $n$ is a positive integer. Let $B$ be the collection of all open intervals $(a,b)$, along with all sets of the form $(a,b)-K$. The topology generated by $B$ is called the $K$-topology on $\mathbb{R}$. When $\mathbb{R}$ is given this topology we denote it by $\mathbb{R}_{K}$. Is $\mathbb{R}_{K}$ a Baire space?
Is $\mathbb{R}_{K}$ a Baire space?
0
$\begingroup$
general-topology
-
0This sounds like a HW exercise from Munkrees. Is this for HW? What have you tried so far? – 2011-05-31
-
0No this is not homework. I just would like to know the general known fact. – 2011-05-31
-
0In my experience, the $K$-topology on $\mathbb{R}$ is useful mainly for providing counterexamples to various statements and to help one navigate the plethora of definitions in point-set topology. In other words, the $K$-topology is a learning tool. So, the best thing to do is to start working through the definitions yourself. If you've already started this process, describe what you've tried and where it has taken you. – 2011-05-31
-
0Actually I think that the topology is the one generated by $B$ and not by $K$. – 2011-05-31