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Martin Sleziak
Martin Sleziak
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In the article Constructions and Applications of Rigid Spaces, IConstructions and Applications of Rigid Spaces, I (Advances in Mathematics 29, 89--130 (1978), V. Kannan and M. Rajagopalan show that if $(2^{\aleph_0})^+ < 2^{2^{\aleph_0}}$, there is a countable space $X$ such that the only non-constant continous self-map is the identity. (Of course that's a notion of rigidity that is "more rigid" than what is asked in the original post.)

In the article Constructions and Applications of Rigid Spaces, I (Advances in Mathematics 29, 89--130 (1978), V. Kannan and M. Rajagopalan show that if $(2^{\aleph_0})^+ < 2^{2^{\aleph_0}}$, there is a countable space $X$ such that the only non-constant continous self-map is the identity. (Of course that's a notion of rigidity that is "more rigid" than what is asked in the original post.)

In the article Constructions and Applications of Rigid Spaces, I (Advances in Mathematics 29, 89--130 (1978), V. Kannan and M. Rajagopalan show that if $(2^{\aleph_0})^+ < 2^{2^{\aleph_0}}$, there is a countable space $X$ such that the only non-constant continous self-map is the identity. (Of course that's a notion of rigidity that is "more rigid" than what is asked in the original post.)

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Dominic van der Zypen
Dominic van der Zypen
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In the article Constructions and Applications of Rigid Spaces, I (Advances in Mathematics 29, 89--130 (1978), V. Kannan and M. Rajagopalan show that if $(2^{\aleph_0})^+ < 2^{2^{\aleph_0}}$, there is a countable space $X$ such that the only non-constant continous self-map is the identity. (Of course that's a notion of rigidity that is "more rigid" than what is asked in the original post.)