Pre-topological space

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2010 Mathematics Subject Classification: Primary: 54A05 [MSN][ZBL]

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.


[1] N.M. Martin, S. Pollard, "Closure spaces and logic" , Kluwer Acad. Publ. (1996)
[2] J.L. Kelley, "General topology" , Graduate Texts in Mathematics 27 Springer (1975) ISBN 0-387-90125-6 Zbl 0306.54002
[3] D. Dikranjan, W. Tholin, "Categorical structures of closure operators" , Kluwer Acad. Publ. (1996)
[4] Jürgen Jost, "Mathematical Concepts", Springer (2015) ISBN 331920436X
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