When proximal continuity and (topological) continuity are the same? Under which conditions proximal continuity of $f$ (having $X\mathrel{\delta_1}Y \Rightarrow f[X]\mathrel{\delta_2}f[Y]$ for every sets $X$, $Y$ on the first proximity) from a proximity $\delta_1$ to a proximity $\delta_2$ and continuity of $f$ for induced topological spaces coincide?
Please with a proof.
 A: According to Wiki article which your referenced “proximity maps will be continuous between the induced topologies”. From the other hand, let $X$ and $Y$ be Hausdorff spaces, $X$ be compact, and $f:X\to Y$ be a continuous map. Then there is a unique proximity $\delta_1$ on the space $X$ whose corresponding topology is the given topology: $A$ is near $B$ if and only if their closures intersect. Since the space $Y$ is compact as a continuous image of a compact space, the proximity $\delta_2$ is also unique an can be defined in the similar manner. Now assume that $A$ and $B$ are arbitrary subsets of the space $X$  such that $A\delta_1B$. Then there exits a point $x\in\overline{A}\cap\overline{B}$. Then $f(x)\in f(\overline{A})\cap f(\overline{B})$. Since the map $f$ is continuous $ f(\overline{A})\subset\overline{f(A)}$ and 
$f(\overline{B})\subset\overline{f(B)}$. Thus $f(x)\in \overline{f(A)}\cap \overline{f(B)}$, so $f(A)\delta_2 f(B)$. Hence the map $f$ is proximal.
I know almost nothing about proximity spaces, but I expect that is general there are easy counterexamples for non-compact $X$, which may be constructed as follows. Let $X$ be any topological space whose topology can be defined by two different proximities $\delta_1$ and $\delta_2$. Then the identity map $i:X\to X$ is continuous, but if both maps $i:(X,\delta_1)\to (X,\delta_2)$ and 
$i^{-1}:(X,\delta_2)\to (X,\delta_1)$ are proximal then $\delta_1=\delta_2$, a contradiction.
