$f : \mathbb R → X$ be a continuous function with respect to the Euclidean topology on $\mathbb R$ and the discrete topology on $X$.

Question

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

Can you please help me out with this? I just started a course on topology and I don't know how to prove this. I know that in a discrete topology, $x_1,x_2$,... Converge to $x$ so $x=x_n$ as {$x$} is open.

Continuous image of a connected set is connected https://math.stackexchange.com/questions/1573795/proof-of-the-continuous-image-of-a-connected-set-is-connected — Sumanta, Sep 15 '20 at 14:00
And connected components of discrete topology are singletons https://math.stackexchange.com/questions/2344396/connected-components-of-a-space-are-a-singletons — Sumanta, Sep 15 '20 at 14:00
Does this answer your question? Describe all continuous function from the reals to the discrete metric space — Chrystomath, Sep 15 '20 at 18:50

score 3 · Answer 1 · answered Sep 15 '20 at 13:59

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HINT: Show that $f^{-1}(\{x\})$ is open and closed for any $x\in X$.

answered Sep 15 '20 at 13:59

checkmath

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The function is continuous by hypothesis, after you that $\mathbb{R}$ is connected with its usual topology. – checkmath Sep 15 '20 at 21:05
How to show that $f^−1({x})$ is open and closed for any $x∈X$? and how does this tell me that $f$ is constant? – JOJO Sep 16 '20 at 10:14
You can prove it sequentially. Since $\mathbb{R}$ is connected we have that if $A$ is open and closed it should be empty or $\mathbb{R}$. – checkmath Sep 16 '20 at 17:42

$f : \mathbb R → X$ be a continuous function with respect to the Euclidean topology on $\mathbb R$ and the discrete topology on $X$.

1 Answers1