Let $X, Y$ be topological spaces.
I want to understand better the structure of the space $C(X,Y)$ of all continuous functions from $X$ to $Y$. Clearly, if $X$ has the indiscrete topology and $Y$ has the discrete topology, then the only continuous functions are the constants.
Now come my questions:
1.) If $X$ is not indiscrete, is there always a non-constant continuous function?
2.) If $Y$ is not discrete, is there always a non-constant continuous function?
3.) If $X$ is not indiscrete and $Y$ is not discrete, is there always a non-constant continuous function?
It seems to be very hard to construct a non-constant continuous map just by knowing that there is one non-trivial open set in $X$ and/or one set in $Y$ not being open. But on the other hand I am not able to construct a counterexample.
Thanks in advance for all help!
Best Answer
Let $X$ be topology on $\{1, 2, 3\}$ with base of $\{1\}, \{1, 2\}, \{1, 3\}$ and $Y$ be any Hausdorff space.
Assume $f(1) \neq f(2)$. Take open $U \subset Y$ s.t. $f(2) \in U$, $f(1) \notin U$. Then $2 \in f^{-1}(U)$, $1 \notin f^{-1}(U)$, so $f^{-1}(U)$ is not open. So $f(1) = f(2)$.
Analogously, $f(1) = f(3)$, and so $f$ is constant.