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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Sign up to join this communityIs 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 $\varphi\restriction_A = f$.
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.