For which spaces are all continuous functions either constant or the identity?
The Sierpinski space has a particular property: All continuous functions to itself are either the identity or constant. Obviously the empty space and the singleton space share this property.
My question is now: Are there other such spaces, and how would you find/construct them?
I already found a few conditions that spaces must fulfil to have this property. Note that in the following, whenever I specify a map, any points not explicitly mentioned are supposed to be mapped to themselves.
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Any space with the property must be a $T_0$ space
If there are two indistinguishable points, the function mapping each of them to the other is continuous.
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The space must be connected.
Otherwise, let $U$ and $V$ be two connected components, and $u\in U$ and $v\in V$ two arbitrarily chosen points in them. Then the map that maps all points of $U$ to $v$, and all points of $V$ to $u$ is continuous.
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Any point contained in an open proper subset $U$ of the space must also be contained in a proper open subset of $U$.
Otherwise, be $p\in U$ not contained in any proper subset of $U$ (note that there can be only one such point, as otherwise those points would be topologically indistinguishable). Then the function that maps the complement of $U$ to $p$ is continuous: Any open set that contains that point necessarily contains all of $U$, and thus its preimage is the full space and thus open; the preimage of any open set that doesn't contain $p$ is the intersection of that set with $U$ and thus open.
Obviously those are by far not sufficient to characterise spaces with the desired property. In particular, all connected metric spaces fulfil all those conditions, but have lots of non-identity non-constant continuous maps.
Edit: another fact about such spaces I've found out: There must not be an open singleton if it is not either the singleton space or the Sierpinski space. To see this, consider the boundary of that open singleton. If it is empty, the open singleton is an isolated point and thus unless it's the singleton space, the space is disconnected. Otherwise, choose a point in the boundary. Then the map that maps the open singleton to itself and all other points to the chosen boundary point is continuous, but clearly not constant and, except in the Sierpinski space, also not the identity.
This also means that any other spaces with that property must be infinite since every finite non-empty $T_0$ space has at least one open singleton.
1 answer
All but one of the spaces under consideration are $T_1$
This is not a complete answer, but note that one can straightforwardly prove that the space is either the Sierpiński space or a $T_1$ space! (We know it is $T_0$ by the argumentation in the OP---if it weren't $T_0$, then swapping two topologically equivalent points would give us a non-identity non-constant continuous endofunction.)
To prove this we shall find a non-constant continuous endofunction for any $T_0$ and non-$T_1$ space, then we shall prove that this endofunction can only be the identity in one case—that of the Sierpiński space.
We know that any non-$T_1$ space contains distinct points $x$, $y$ such that $x \in \text{cl}(\{y\})$ (so $\{x,y\}$ is homeomorphic to the Sierpiński space, given that our space is $T_0$). But any nontrivial $T_0$ space has a nontrivial continuous map to the Sierpiński space. Specifically, let $O$ be any nontrivial open and $C$ be its complement; then the map that takes $O$ to $y$ and $C$ to $x$ is continuous and non-constant!
(And we see that, in the case of the Sierpiński space, this map is equal to the identity; in all other cases, the map cannot be equal to the identity.)
As for the more general question I suspect that it depends on the Axiom of Choice, and more particularly that, in certain models of ZF, there are subsets of the reals which have no endomorphisms except constant functions and the identity; but I am not sure of this.
It may also be useful to note that, because (as the OP noticed) the space must be connected, it must be either uncountable or non-regular, as all countable regular spaces are metrizable and hence disconnected.
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