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.
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