Showing a continuous function between two topologies is constant

Question

Let $X$ be an arbitrary set, and let $f: \Bbb R → X$ be a function that is continuous with respect to the euclidean topology on $\Bbb R$ and the discrete topology on $X$. Prove that $f$ is constant.

So basically, in a different scenario, we're given the structure of the topological basis we're working with. In my case, however, I'm not very sure as to how to approach this in a more general way, without being given a basis to work with. Any ideas?

Continuous image of a connected set is connected and connected components of discrete topology are singletons. I think now you can do it. — Sumanta, Sep 17 '20 at 14:28

score 1 · Answer 1 · answered Sep 17 '20 at 14:37

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Since $f$ is continuos and $\mathbb{R}$ is connected $f(\mathbb{R})$ is connected, but a connected in a discrete space are the singletons, hence $f$ is costant.

To see that a connected in a discrete space are a singleton you can proceed as follows : take $x \in X$ since $X$ has the discrete topology $x$ is clopen, but $\left\lbrace x \right\rbrace \in C(x)$, hence $\left\lbrace x \right\rbrace = C(x)$.

answered Sep 17 '20 at 14:37

jacopoburelli

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I understand now, thank you for your help. – Dwayne Johnson Sep 17 '20 at 16:46

Showing a continuous function between two topologies is constant

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