Some new separation axioms are introduced and studied. We also deal with maps having an extension to a homeomorphism between the Wallman compactifications of their domains and ranges.

A function $f:X\rightarrow X$ determines a topology $P(f)$ on $X$ by taking the closed sets to be those sets $A\subseteq X$ with $f(A)\subseteq A$ . The topological space $(X,P(f))$ is called a functionally Alexandroff space. We completely characterise the homogeneous functionally Alexandroff spaces.

Alexandroff topologies play an enigmatic role in topology. An important family of Alexandroff topologies are the functional Alexandroff spaces introduced by Shirazi and Golestani, and called primal...

If $f:X\to X$ is a map, then the family $\{A\subseteq X:f(A)\subseteq A\}$ gives the closed sets of a topology P(f) on X called a functionally Alexandroff topology. If $(X,P(f))$ is a connected homogeneous functionally Alexandroff space, we give an algebraic characterization and some basic properties of its autohomeomorphism group $\mathcal{H}(X)$.

A topological space is called resolvable if it is a union of two disjoint dense subsets, and is n-resolvable if it is a union of n mutually disjoint dense subsets. Clearly a resolvable space has no isolated points. If f is a selfmap on X, the sets A⊆X with f (A)⊆A are the closed sets of an Alexandroff topology called the primal topology