The cofinite topology on $\mathbb{R}$ is the topology in which a subset $F⊆\mathbb{R}$ is closed if and only if $F$ is either finite or $F = \mathbb{R}$. Let $X = \mathbb{R}$ with the cofinite topology and $Y = \mathbb{R}$ with the usual topology. Show that any continuous map $f : X →Y$ is a constant.
I was trying to solve this problem by contradiction but could not proceed much. Can anyone help me please to tackle this problem?
Best Answer
HINT: Suppose that $f:X\to Y$ is not constant; then there are $x_0,x_1\in X$ such that $f(x_0)\ne f(x_1)$. Let $U_0$ and $U_1$ be disjoint open nbhds of $f(x_0)$ and $f(x_1)$, respectively. Then $f^{-1}[U_0]$ and $f^{-1}[U_1]$ are open nbhds of $x_0$ and $x_1$ in $X$. Use the fact that $U_0\cap U_1=\varnothing$ to get a contradiction.