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Is there a connected $T_2$-space $(X,\tau)$ with $|X|>1$ and the following property?

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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    $\begingroup$ There is no such metric space: note a connected metric space is uncountable, and we can pick as $A$ some nontrivial convergent sequence together with its limit. Then an autohomeomorphism can't swap the limit with some other point of the sequence while keeping all its other points in place. $\endgroup$
    – Wojowu
    Oct 14 '20 at 14:04
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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) 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, ZBL07224267. ArXiv version

[2] Bing, R. H., A connected countable Hausdorff space, Proc. Am. Math. Soc. 4, 474 (1953). ZBL0051.13902.

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