Topology as Magma of Sets

Theorem

The concept of a topology is an instance of a magma of sets.

Proof

It will suffice to define partial mappings such that the axiom for a magma of sets crystallises into the axioms for a topology.

Let $X$ be any set, and let $\powerset X$ be its power set.

Define:

$\phi_1: \powerset X \to \powerset X: \map {\phi_1} S := X$

$\phi_2: \powerset X^2 \to \powerset X: \map {\phi_2} {S, T} := S \cap T$

For each index set $I$, define:

$\ds \phi_I: \powerset X^I \to \powerset X: \map {\phi_I} {\family {S_i}_{i \mathop \in I} } := \bigcup_{i \mathop \in I} S_i$

It is blatantly obvious that these partial mappings capture the axioms for a topology.

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$\blacksquare$