Let $\{A_\alpha\}$ be a collection of subsets of some topological space $X$ s.t. $X=\cup_{\alpha}A_\alpha.$ Let $f:X\to Y$, where $Y$ is some other topological space. Suppose that $f|A_\alpha$ is continuous for each $\alpha$. Find an example where the collection $\{A_\alpha\}$ is countable and each $A_\alpha$ is closed, but $f$ is not continuous. [Munkres 2/e, 18.9(b)]
i found the answer but i did n't understand the red line .....
Can any body help me understanding the red lines?,,,as im not getting and understand in my head.....
thanks in advance
The only integer contained in $(-1/2, 1/2)$ is $0$, so we have that $f^{-1}(-1/2, 1/2) = \{0\}$, since $f$ is the identity map (or inclusion map). Now $(-1/2, 1/2)$ is open in $\mathbb{R}$ with the standard topology, so if $f$ is continuous, its preimage $\{0\}$ should be open in $\mathbb{Z}$ with the cofinite topology. The open sets in the cofinite topology are by definition the empty set and any subset whose complement is finite. Thus $\{0\}$ is not open in the cofinite topology on $\mathbb{Z}$ since $\mathbb{Z}\setminus \{0\}$ is not finite. Thus $f$ is not continuous.