Let X be a topological space. show that if there exists a continuous, non constant map from X to the integers with the discrete topology, then X is not connected.
So I know that connected subspaces of integers with the discrete topology are just points. Also the image of a connected space under a continuous map is connected.
Here is where my reasoining for the proof eludes me. If I take the inverse image of those points is it that I haven now created a separation in the inverse image thus showing that X is not connected? Or is it that since the image of a connected space under a continuous function is connected, but since this maps to a point, the function is therefore constant?
Best Answer
If you already know that the image of a connected set is connected, then you're done: if $f:X\to \mathbb{Z}$ is connected, then $f(X)$ is connected. The only connected subsets of $\mathbb{Z}$ are points, so $f(X)$ is a point, i.e. $f$ is constant.