Is there a connected $T\_2$-space $(X,\tau)$ with $|X|>1$ and the following property?



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> Whenever $A$ is a subset of $X$ with $|A|<|X|$ and $f:A\to A$ is a bijection, there is a homeomorphism $\varphi:X\to X$ such that $\varphi\restriction\_A = f$.
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Highly homogeneous connected $T_2$-spaces
Bing's connected countable space $\mathbb{B}$ (see [2]) is such an example. Work of Banakh, Banakh, Hryniv, and Stelmakh [1] (motivated by a [MathOverflow question](https://mathoverflow.net/questions/286366/is-bings-countable-connected-space-topologically-homogeneous)) gives you what you want.


Note that they prove that any bijection between two $\theta$-discrete subsets of $\mathbb{B}$ extends to a homeomorphism of $\mathbb{B}$, and it is immediate that every finite subset of $\mathbb{B}$ is $\theta$-discrete.


[1] *Banakh, Iryna; Banakh, Taras; Hryniv, Olena; Stelmakh, Yaryna*, [**The connected countable spaces of Bing and Ritter are topologically homogeneous**](http://topology.auburn.edu/tp/restricted/v57/tp57010.pdf), [ZBL07224267](https://zbmath.org/?q=an:07224267). [ArXiv version](https://arxiv.org/abs/1712.01964)


[2] *Bing, R. H.*, [**A connected countable Hausdorff space**](http://dx.doi.org/10.1090/S0002-9939-1953-0060806-9), Proc. Am. Math. Soc. 4, 474 (1953). [ZBL0051.13902](https://zbmath.org/?q=an:0051.13902).


