TITLE: Homeomorphisms which switch 2 points
QUESTION [5 upvotes]: Let $X$ be a space and suppose: 
For every two points $x,y\in X$ there is a homeomorphism $h$ that maps $X$ onto itself and such that $h(x)=y$ and $h(y)=x$.
Is there a name for  this property?  
Obviously $X$ is homogeneous. Does every homogeneous space have this property?  If not, are there some classes of homogeneous spaces which have the property?  For instance does every homogeneous compact connected metric space?

REPLY [2 votes]: Such spaces are called bi-homogeneous (or perhaps bihomogeneous).
An example of a homogeneous space which is not bi-homogeneous is the long-line without the initial point (i.e., $X = ( \omega_1 \times [0,1) ) \setminus \{ ( 0 , 0 ) \}$).  It is fairly easy to show that $X$ is homoegeneous, but since all homeomorphisms $X \to X$ are order-preserving it cannot be bi-homogeneous.
Many of the homogeneous spaces you encounter "in the wild" will be bi-homogeneous, but there doesn't appear to be many classes of topological spaces where homogeneity implies bi-homogeneity. One class of spaces for which this implication is (non-vacuously) true is the class of topological groups.
Homogeneous but not bi-homogeneous continua have been constructed. For one (and I believe the first) example see 

Kuperberg, Krystyna, On the bihomogeneity problem of Knaster, Trans. Am. Math. Soc. 321, No. 1, 129-143 (1990). ZBL0707.54025.