How to describe all continuous maps from T' to $T$, where $T=\mathbb{R}$ with natural topology (base given by the intervals $(a,b)$ ), and T'=\mathbb{R} with the topology with basis given by the half-intervals $(a,b]$ ?
Obviously, all "linear" maps $F: (a,b] \to (A_1a+A_2, B_1b+B_2)$ will be continuous. (their $F^{-1}$ are, obviously, bijections)
Add:Looks, like if all "ends" of intervals a and b are mapped according to some bijective function, the map is continuous.
Will there be any other?
Edit:replaced relations with maps. Should have looked at the dictionary earlier.