Modal logic and topology
You can't say much about modal logic in general. You have to be more specific to get anywhere. You have to choose some axioms. Ideally the axioms you need for your application correspond to a named set of axioms that has been studied before.
The situation is similar in point-set topology. You can't say very much about a general topological space. You have to specify some separation axioms to get going.
Bare bonesModal logicA modal logic is any set of formulas in the modal language that:
- contains all propositional tautologies,
- is closed under modus ponens, and
- is closed under uniform substitution.
In particular, this definition requires nothing of the modal operator (box"). You just have propositional logic with a funny symbol added that could mean anything.
TopologyA topological space is a setX along with a set of subsets ofX called open sets. The empty set and the full spaceX are open sets. Furthermore, the set of open sets is closed under finite intersections and arbitrary unions.
There's not much you can say about topological spaces in general because, for example, the definition includes extreme cases such as the discrete topology (every subset of X is open) and the indiscrete topology (only the empty set andX are open).
Regular and normalLike many areas of mathematics, logic and topology use the terms regular" and normal" to refer to systems with common choices of extra structure.
Modal logicA regular modal logic is a normal modal logic with a second modal operator (diamond") that satisfies
p ( p)
and has the inference rule(p q) r implies (p q) r.
A modal logic is normal if it satisfies the axiom
(p q) ( p q)
and the inference rule that ifp is a theorem, p is also a theorem.
TopologyTopology also usesregular andnormal to refer to adding a few axioms.
A regular topological space is one in which you can separate points from closed sets. Given a pointx and a closed setF not containingx, there exist disjoint open sets U and V such thatx is contained inU andF is contained inV. [1]
A normal topological space is one in which you can separate disjoint closed sets.
For many mathematicians, a metric space is the weakest topology they're interested in, and metric spaces are normal. But weaker topologies come up. The Zariski topology in algebraic geometry is not regular, and the weak topology on an infinite dimensional Banach space is regular but not normal.
Related posts[1] Why do we useF to denote a closed set? It's a convention that goes back to the French wordferme for closed."
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