If we look at any set $X$ with the trivial topology, then all functions from $X$ to $X$ are continuous. We could also take the discrete topology and get the same result: all functions are continuous.
Another example: Take the Sierpinski space, all functions from it to itself, except the function that switches 0 and 1, are continuous. Again there is another topology that gives the same continuous functions (just take the other singleton as open).
Is this true for $\mathbb{R}$ aswel? Note that I'm not asking if $\mathbb{R}$ is completely regular, which is a seemingly similar but different property (In that property, the image space is equipped with the Euclidean topology if I'm not mistaken).
I'm interested in $\mathbb{R}$ especially because this would give another way to think about continuous functions on it. High school students learn an epsilon delta definition but you can teach them what an open set is and define continuous that way. The usual definition of open is then "union of open intervals", but maybe there is a different choice of opens to get the same notion of continuous functions.
That being said, I'm also interested in other spaces on which the topology can be changed to obtain the same continuous functions to itself, like the examples I gave, or in spaces where you can show that the topology is the only one that gives those continuous maps.
Thanks in advance
edit:
Assume $X$ has more than 1 element. The argument that CarryonSmiling makes can be generalised as follows:
Let's call a space that is T1 and connected a Tc space. If two topologies on $X$ have the same continuous maps from $X$ to itself, then they are either both Tc or both not Tc. This is true because Tc is equivalent to the property that CarryonSmiling uses, that is expressed only with continuous maps from $X$ to itself:
$$(*)\quad\mbox{No map $h:X\rightarrow X$ whose range has exactly two elements is continuous.}$$
To see this equivalency, note that $$T1 \iff \mbox{closed singletons} \iff \mbox{every subspace with exactly 2 elements is discrete}$$
$$\mbox{connected} \iff \mbox{every continuous map to the discrete space with 2 elements is constant}$$
That means that T1 and connectedness together are exactly $(*)$.
Maybe this can help to find a space that is determined by its continuous maps to itself. Please let me know if any part of this reasoning is invalid.
Best Answer
Suppose $\tau$ is such a topology. For clarity, I'll use "open" to mean open in the usual sense, and "$\tau$-open" to mean open in the new topology, $\tau$; similarly with "continuous" and "$\tau$-continuous."
We'll show that $\tau$ is in fact a refinement of the usual topology on $\mathbb{R}$. The only fact about the usual topology used is $$(*)\quad\mbox{No map $h:\mathbb{R}\rightarrow\mathbb{R}$ whose range has exactly two elements is continuous.}$$
We begin by showing that $\tau$ is $T_1$. Take $a\neq b$, and take two non-empty subsets $A,B$ with $A\cap B=\varnothing$ and $A\cup B=\mathbb R$ and with $B$ $\tau$-open. Such $A, B$ exist since $\tau$ is not indiscrete: if $\tau$ were, it would have too many continuous functions.
Now by $(*)$, the function that sends $A$ to $a$ and $B$ to $b$ is not continuous in the usual topology, so it is not $\tau$-continuous either.
Since $f$ isn't $\tau$-continuous, we must have that $A$ is not $\tau$-open. Now, the only $f$-preimages are $\emptyset, B, \mathbb{R}, A$; the only one of these which is not $\tau$-open is $A$, so for $f$ to not be $\tau$-continuous there must be some $\tau$-open set $U$ with $f^{-1}(U)=A$. That is, there must be a $\tau$-open set that contains $a$ and not $b$.
So we have that $\tau$ is $T_1$. (Note: $\tau$ is also connected by the same argument, but that's not directly useful for the rest of this argument.)
We can now prove that $\tau$ refines the usual topology. Since $\tau$ is $T_1$, we know that $\mathbb R\setminus \{x\}$ is $\tau$-open for all $x$. For $a\in\mathbb{R}$, consider the function $f_a(x)=\max(a,x)$. This function is continuous and hence $\tau$-continuous. Since $\mathbb R\setminus \{a\}$ is $\tau$-open, its $f_a$-preimage must be $\tau$-open; that is, $(a,\infty)$ is $\tau$-open. We can prove $(-\infty,a)$ is open analogously. Intersecting open sets, we get that $(a, b)$ is $\tau$-open for every $a,b\in\mathbb{R}$.