homotopy between $\mathbb{R}$ and a discrete space

Question

Any two continuous functions from the real line $\mathbb{R}$ to any discrete space $Y$ are

$(a)$ never homotopic.

$(b)$ homotopic if one of them is a constant function.

$(c)$ homotopic if both are constant functions.

$(d)$ always homotopic.

I don't really know much about concepts of algebraic topology. All I know is that a homotopy of two continuous functions $f,g$ on $\mathbb{R}$ to some discrete space $Y$ is defined as the continuous map $H:\mathbb{R}\times [0,1]\to Y$ such that $H(x,0)=f(x)$ and $H(x,1)=g(x)$ for all $x\in \mathbb{R}$. On the other hand, a homotopy between two spaces may be defined as a continuous deformation of a space curve on the domain space to the range space. I could only make a choice of option $(a)$ as I think there is no possibility of having a "continuous" deformation to some "discrete" space. But this is not a satisfying logic to me. Can someone give a concrete mathematical proof of this concept? Any help is appreciated.

Answer 1

Note first that if $f: \Bbb R \to Y$ is continuous (and $Y$ is discrete) then $f[\Bbb R]$ is non-empty connected in $Y$ and so a singleton (!) and $f$ must be a constant map.

So even without mentioning homotopies, we are only talking about the set of constant functions from $\Bbb R$ to $Y$ anyway.

Also, a homotopy $H$ between $f$ and $g$ is a (special) continuous map from $\Bbb R \times [0,1]$, which is also connected, so that the same argument applies and

$$\exists y_0 \in Y: \forall x \in \Bbb R : \forall t \in [0,1]: H(x,t) = y_0$$

as the image of $H$ is a singleton by connectedness.

Now as $H(x,0)=f(x)$ for all $x \in \Bbb R$ (one extra condition of a homotopy), $f$ is the constant map with value $y_0$ and similarly $g$ is too, as $g(x)=H(x,1)$ for all $x \in \Bbb R$. So $f=g$ and both are the same constant function!

Now what answer is correct? This is partly a language interpretation question.

(a) is correct if you (implicitly) assume that "any two continuous functions" in the question means any two different continuous functions. We've seen two functions are only homotopic here if they're the same.

(b) is false, take $f\equiv 0$ and $g \equiv 1$. One of them is constant (in fact both of them) but they're not homotopic.

(c) is false by the same example. Both are constant functions etc.

(d) is certainly false.

So what remains is (a) under the mentioned implicit assumptions that the two implies distinctness.