Is there an infinite Hausdorff space $(X,\tau)$ with the following property?

If $x\neq y \in X$ and $f:\{x,y\}\to X$ is a map, then there is exactly one continuous function $f': X\to X$ such that $f'|_{\{x,y\}} = f$.

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    $\begingroup$ A few observations: (1) By uniqueness, the maps $f'$ are forced to be either homeomorphisms or constant maps, leading to the question whether spaces exists admitting only maps of this form. (2) If $X$ is (locally?) compact then $X$ cannot have non-trivial path-components by an application of Tietze's Extension Theorem. (3) The space $X$ will satisfy curious properties such as $\operatorname{Hom}(X \times X,X) \cong \operatorname{Hom}(X \sqcup X,X)$. $\endgroup$ – Sofie Verbeek Sep 12 '18 at 15:09
  • $\begingroup$ @SofieVerbeek: Concerning your first observation, let me point out that such spaces do exist: see mathoverflow.net/questions/294511/…. $\endgroup$ – Will Brian Sep 12 '18 at 16:16

Using the technique of van Mill it is possible to prove the following

Theorem. There exists a subset $Z$ in the complex plane $\mathbb C$ such that

1) for any complex numbers $x,a,b,c,d\in Z$ with $a\ne b$ and $c\ne d$ we have $h_{a,b,c,d}(Z)\subset Z$ where $h_{a,b,c,d}(z):=c+\frac{d-c}{b-a}(z-a)$;

2) for any homeomorphism $h:Z\to Z$ there exist unique complex numbers $a,b,c,d\in Z$ with $a\ne b$ and $c\ne d$ such that $h=h_{a,b,c,d}$;

3) Under CH (more generally, under $\acute{\mathfrak n}=\mathfrak c$) for any non-constant continuous map $f:Z\to Z$ there exist unique points $a,b,c,d\in Z$ with $a\ne b$ and $c\ne d$ such that $f=h_{a,b,c,d}$.

Here by $\acute{\mathfrak n}$ denotes the smallest cardinality of an infinite cover of $[0,1]$ by pairwise disjoint closed sets. By the classical Sierpi$\acute{\mathfrak n}$ski Theorem, $\aleph_1\le \acute{\mathfrak n}\le\mathfrak c$. More information on the small uncountable cardinal $\acute{\mathfrak n}$ can be found in this MO-post.

Remark. Under $\acute{\mathfrak n}=\mathfrak c$, the space $Z$ from Theorem has the property required in the question of Dominic van der Zypen: for any distinct points $a,b\in Z$ and point $c,d\in Z$ the map $h_{a,b,c,d}$ is a unique continuous self-map $f:Z\to Z$ such that $f(a)=c$ and $f(b)=d$.

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    $\begingroup$ I love how there's CH lurking around the corner! $\endgroup$ – Sofie Verbeek Sep 13 '18 at 7:19
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    $\begingroup$ Incredible, that you came up with this answer, and as @SofieVerbeek says, I would never have thought that CH comes into the picture! $\endgroup$ – Dominic van der Zypen Sep 13 '18 at 7:31
  • $\begingroup$ @SofieVerbeek I admit that there exists also a ZFC-example, constructed with the help of the Cook continuum, but the construction has to be more complicated: inductively we should glue Cook continua by pairs of points and then iterate this construction, etc. $\endgroup$ – Taras Banakh Sep 13 '18 at 7:53

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