I am well aware that $(X,\tau)$ is called a topological space iff
- $X\in\tau,\emptyset\in\tau$
- $\forall\alpha\in I,U_{\alpha}\in\tau$ implies $\bigcup_{\alpha\in I}U_a\in\tau$
- $V\in\tau,W\in\tau$ implies $V\cap W\in\tau$
Similarly, I am well aware that $(X,\mathcal F)$ is called a field of sets iff
- $X\in\mathcal F$
- $S\in\mathcal F$ implies $X\setminus S\in\mathcal F$
- $V\in\mathcal F,W\in\mathcal F$ implies $V\cap W\in\mathcal F$
with axioms 1 and 2 of the latter implying
- $\emptyset\in\mathcal F$
and axioms 2 and 3 of the latter implying
- $V\in\mathcal F,W\in\mathcal F$ implies $V\cup W\in\mathcal F$.
I find this rather curious, because for topological spaces, axiom 1 coincides with axiom 1 and proposition 4 combined for fields of sets, and for both spaces, axiom 3 is identical, not to mention that axiom 2 for topological spaces is just a stronger version of proposition 5 for fields of sets. The only axiom for which a non-trivial distinction exists is in axiom 2 for fields of sets, which involves the relative complement, which shows up nowhere in the axioms for a topological space,... BUT, if a topological space also satisfies axiom 3 for fields of sets, then every open set in the space is actually closed as well (and thus, clopen).
As such, I get the impression that topological spaces and fields of sets are both special cases of a more general and possibly more fundamental type of mathematical space; alternatively, if they are not special cases of a more fundamental type of mathematical space, then it intuitively seems as though a deeper connection between the two types of spaces should exist.
With this context provided, my two questions are:
- Does there exist a named and rigorously studied mathematical structure/space that generalizes/unifies both ideas, as my intuition suggests? If so, then what are some sources on the topic/theory?
- If the answer to the above question is no, then instead, is there some type of fundamental connection between the two types of spaces, and what is that connection, if it exists? Again, sources would be appreciated here, even a Wikipedia article would do it for me.
