Set-theoretic topology
It focuses on topological questions that can be solved using set-theoretic methods, for example, Suslin's problem.
Zoltán Balogh gave the first ZFC construction[2] of a small (cardinality continuum) example, which was more well-behaved than Rudin's.
Using PCF theory, M. Kojman and S. Shelah constructed[3] a subspace of Rudin's Dowker space of cardinality
The answer to the normal Moore space question was eventually proved to be independent of ZFC.
For any cardinal k, we define a statement, denoted by MA(k): For any partial order P satisfying the countable chain condition (hereafter ccc) and any family D of dense sets in P such that |D| ≤ k, there is a filter F on P such that F ∩ d is non-empty for every d in D.Since it is a theorem of ZFC that MA(c) fails, Martin's axiom is stated as: Martin's axiom (MA): For every k < c, MA(k) holds.In this case (for application of ccc), an antichain is a subset A of P such that any two distinct members of A are incompatible (two elements are said to be compatible if there exists a common element below both of them in the partial order).
) is false: [0, 1] is a compact Hausdorff space, which is separable and so ccc.
An equivalent formulation is: If X is a compact Hausdorff topological space which satisfies the ccc then X is not the union of k or fewer nowhere dense subsets.
Martin's axiom has a number of other interesting combinatorial, analytic and topological consequences: Forcing is a technique invented by Paul Cohen for proving consistency and independence results.
It was first used, in 1963, to prove the independence of the axiom of choice and the continuum hypothesis from Zermelo–Fraenkel set theory.
Forcing was considerably reworked and simplified in the 1960s, and has proven to be an extremely powerful technique both within set theory and in areas of mathematical logic such as recursion theory.
, and then introduce an expanded membership relation involving the "new" sets of the form
Forcing is a more elaborate version of this idea, reducing the expansion to the existence of one new set, and allowing for fine control over the properties of the expanded universe.
