Phase and Proximity

When considering “distance” as a concept to express “proximity”, it is implicitly assumed that the distance between two points corresponds to a real number value. The fact that it is a real number implicitly assumes universal sequential. However, there are use cases where the whole order is not necessarily necessary for the proximity argument.

• In topology, “closer” is semi-sequential (e.g. equations) since “there is an open set in implication, which is an element of the inner open set”.

Definition of phase Let X be a set and $\mathcal{O}$ be a subset of the power set $\mathfrak{P}(X)$. When \mathcal{O}} satisfies the following property, the pair $(X,\mathcal{O})$ is called a topological space with X as the platform set and \mathcal{O}} as the open set system, and the source of $\mathcal{O}$ is called an open set in X.

1. $\varnothing ,X\in \mathcal{O}$
2. $\forall O_1,O_2\in \mathcal{O}:O_1\cap O_2\in \mathcal{O}$
3. $\forall \{O_{\lambda }{\}}_{\lambda \in \Lambda }\subset \mathcal{O}:{\bigcup }_{\lambda \in \Lambda }{O}_{\lambda }\in \mathcal{O}$

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