Let $\kappa>0$ be a cardinal and let $(X,\tau)$ be a topological space. We say that $X$ is $\kappa$-homogeneous if

1. $|X| \geq \kappa$, and
2. whenever $A,B\subseteq X$ are subsets with $|A|=|B|=\kappa$ and $\psi:A\to B$ is a bijective map, then there is a homeomorphism $\varphi: X\to X$ such that $\varphi|_A = \psi$.

Questions: Is it true that for $0<\alpha < \beta$ there is a space that is $\beta$-homogeneous, but not $\alpha$-homogeneous? Is there even such a space that is $T_2$? Also it would be nice to see an example for $\alpha=1, \beta=2$. And I was wondering whether there is a standard name for $\kappa$-homogeneous spaces. (Not all of these questions have to be answered for acceptance of answer.)

Let $\kappa>0$ be a cardinal and let $(X,\tau)$ be a topological space. We say that $X$ is $\kappa$-homogeneous if

1. $|X| \geq \kappa$, and
2. whenever $A,B\subseteq X$ are subsets with $|A|=|B|=\kappa$ and $\psi:A\to B$ is a bijective map, then there is a homeomorphism $\varphi: X\to X$ such that $\varphi|_A = \psi$.

Questions: Is it true that for $0<\alpha < \beta$ there is a space $X$ such that $|X|\geq \beta$, and $X$ is $\alpha$-homogeneous, but not $\beta$-homogeneous? Is there even such a space that is $T_2$? Also it would be nice to see an example for $\alpha=1, \beta=2$. And I was wondering whether there is a standard name for $\kappa$-homogeneous spaces. (Not all of these questions have to be answered for acceptance of answer.)