nLab unified topological space

Idea

A notion of topological space which comes with predefined notions of neighbourhood, apartness, and nearness, without having to define any of them in terms of the others.

Definition

A unified topological space is a set $A$ with a unified topology - three relations $\ll$, $\bowtie$, and $\approx$ between $A$ and its power set $\mathcal{P}(A)$, such that for all elements $x$ of $A$ and subsets $U$ and $V$ of $A$,

• $\ll$ is a topology in the usual sense:

  • if $x \ll U$, then $x \in U$

  • if $x \ll U$ and $U \subseteq V$, then $x \ll V$

  • $x \ll A$

  • if $x \ll U$ and $x \ll V$, then $x \ll U \cap V$

  • if $x \ll U$, then $x \ll \{y \in A \vert y \ll U\}$

• $\bowtie$ satisfies the following apartness axioms:

  • if $x \bowtie U$, then $x \in U$ is false

  • if $x \bowtie U$ and $U \subseteq V$, then $x \bowtie V$

  • $x \bowtie \emptyset$

  • if $x \bowtie U$ and $x \bowtie V$, then $x \bowtie U \cup V$

  • if $x \bowtie U$, then $x \bowtie \{y \in A \vert y \approx U\}$

• $\approx$ satisfies the following closure space axioms:

  • if $x \in U$, then $x \approx U$

  • if $x \approx U$ and $U \subseteq V$, then $x \approx V$

  • $x \approx \emptyset$ is false

  • if $x \approx U \cup V$ and $x \approx U$, then $x \approx V$

  • if $x \approx \{y \in A \vert y \approx U\}$, then $x \approx U$

• The following compatibility condition holds:

  • if $x \ll U$ and $x \approx V$, then there exists an element $y$ of $A$ such that $y \in U \cap V$.

 Related concepts

• topological space

• antithesis interpretation

 References

Unified topologies were defined in definition 10.17 of:

• Michael Shulman, Affine logic for constructive mathematics. Bulletin of Symbolic Logic, Volume 28, Issue 3, September 2022. pp. 327 - 386 (doi:10.1017/bsl.2022.28, arXiv:1805.07518.