# Highly homogeneous connected $T_2$-spaces

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 $$\varphi\restriction_A = f$$.

• 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. Oct 14 '20 at 14:04

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.