View source for Pre-topological space

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Let $X$ be a set and $\mathcal{P}X$ the set of subsets of $X$.  A pre-topological space structure on $X$ is defined by a ''Čech closure operator'', a mapping $C : \mathcal{P}X \rightarrow \mathcal{P}X$ such that

C1) $C(\emptyset) = \emptyset$;

C2) $A \subseteq C(A)$;

C3) $C(A \cup B) = C(A) \cup C(B)$. 

A set $A$ in $X$ is ''closed'' if $A = C(A)$.

A mapping between pre-topological spaces $f : X \rightarrow Y$ is ''continuous'' if $f(C_X(B)) \subseteq C_Y(f(B))$ for any $B \subseteq C$.

If the operator $C$ also satisfies (C4) $C(C(A)) = C(A)$, then $X$ is a [[topological space]] with $C$ as the [[Kuratowski closure operator]].  

==References==
<table>
<TR><TD valign="top">[1]</TD> <TD valign="top"> N.M. Martin, S. Pollard,   "Closure spaces and logic" , Kluwer Acad. Publ.  (1996)</TD></TR>
<TR><TD valign="top">[2]</TD> <TD valign="top"> J.L. Kelley, "General topology" , Graduate Texts in Mathematics '''27''' Springer  (1975) {{ISBN|0-387-90125-6}} {{ZBL|0306.54002}}</TD></TR>
<TR><TD valign="top">[3]</TD> <TD valign="top"> D. Dikranjan, W. Tholin, "Categorical structures of closure operators" , Kluwer Acad. Publ.  (1996)</TD></TR>
<TR><TD valign="top">[4]</TD> <TD valign="top"> Jürgen Jost, "Mathematical Concepts", Springer (2015) {{ISBN|331920436X}}</TD></TR>
</table>