Does there exist a topological space Y containing more than one point such that any function $f : X\to Y$ is continuous?

Question

I found this question in one of my past question papers of the college.

Let $X$ be any topological space. Does there exist a topological space $Y$ containing more than one point such that any function $f : X\to Y$ is continuous? Prove your claim or give an example of such a topological space.

I feel like this should not be true but I am not able to think of a counterexample. Any help would be appreciated

Answer 1

Let $Y$ be any set, and give it the topology in which the only two open subsets of $Y$ are $\emptyset$ and $Y$. For any $f: X \to Y$, we see that $f^{-1}(\emptyset) = \emptyset$ and $f^{-1}(Y) = X$. These are both open in $X$, so that $f$ is continuous. This worked for arbitrary functions $f: X \to Y$.

Answer 2

The statement is true, hence it is unspurprising that you could not prove it wrong.

In fact, for every set $Y$ there is a topology $\tau$ on $Y$ such that for every topological space $\mathbf{X}$ all functions $f : \mathbf{X} \to (Y,\tau)$ are continuous.

To prove this, take note that continuity does not care about points at all, but rather about open sets. In particular, the potential obstacles to $f : \mathbf{X} \to (Y,\tau)$ are all coming from the open sets $U \in \tau$.