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 $\mathcal P \left({X}\right)$ be its power set.

Define:


 * $\phi_1: \mathcal P \left({X}\right) \to \mathcal P \left({X}\right): \phi_1 \left({S}\right) := X$


 * $\phi_2: \mathcal P \left({X}\right)^2 \to \mathcal P \left({X}\right): \phi_2 \left({S, T}\right) := S \cap T$

For each index set $I$, define:


 * $\phi_I: \mathcal P \left({X}\right)^I \to \mathcal P \left({X}\right): \phi_I \left({\left({S_i}\right)_{i \in I}}\right) := \displaystyle \bigcup_{i \mathop \in I} S_i$

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